Systems and processes for building multiple equiprobable coherent geometrical models of the subsurface

ABSTRACT

A method, apparatus and system for, in a computing system, perturbing an initial three-dimensional (3D) geological model using a 3D vector field. A coherent 3D vector field including 3D vectors may be generated where each 3D vector of the 3D vector field is associated with a node of the initial 3D geological model and has a magnitude within a range of uncertainty of the node of the initial 3D geological model associated therewith. The coherent 3D vector field may be applied to the initial 3D geological model associated therewith to generate an perturbed 3D model. The perturbed 3D model may differ from the initial 3D geological model by a displacement defined by the 3D vector field associated with nodes having uncertain values. The perturbed 3D model may be displayed.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation application of U.S. patentapplication Ser. No. 12/791,370, filed Jun. 1, 2010, now U.S. Pat. No.8,600,708, which in turn claims the benefit of prior U.S. ProvisionalPatent Application No. 61/182,804, filed Jun. 1, 2009, both of which areincorporated by reference herein in their entirety.

FIELD OF THE INVENTION

The invention pertains to the general field of modeling stratifiedterrains in the subsurface.

More precisely, new techniques are proposed for the purpose of modelingthe geological horizons and the fault network. Contrary to othermethods, the proposed new methods allow the geological consistency ofthe geometry of these horizons and faults to be guaranteed. Moreover,the proposed modeling technique allows the geometrical uncertainties tobe taken into account and to stochastically generate a plurality ofequiprobable models honoring coherency constraints.

BACKGROUND OF THE INVENTION

There are many sources of uncertainties in modeling geological data.Uncertainties may include, for example, the following:

-   -   The location of each point in a subsurface terrain may have an        uncertainty approximately proportional to the wavelength of the        received seismic signals reflected from the subsurface point.        Since the wavelengths of recorded seismic signals are often        approximately 30 meters, the magnitude of the uncertainty of        each subsurface point in such a case may be approximately 60        meters, a significant value.    -   The locations of sampling points on horizons and faults may be        derived based on an estimate of seismic velocities. In practice,        seismic velocities may only be known approximately. As a        consequence, the locations of the sampling points derived        therefrom are uncertain.    -   A particular type of sampling point on horizons called “well        markers” is located at the intersections of well paths and        horizons or faults. However, the geometry of the well path that        defines these well markers may only be known approximately. As a        consequence, the location of well markers is uncertain, which in        turn causes the location of the horizons intersected thereby to        be uncertain.

When seismic data is uncertain, a model computed using the data may alsobe uncertain.

There is a great need in the art for a mechanism to accurately modelseismic data, which itself includes inherently inaccurate information.

SUMMARY OF THE INVENTION

Embodiments of the invention provide a system and method for modelingstratified terrains of a geological subsurface including geologicalhorizons and fault networks. The geological coherency of the geometry ofhorizons and faults may be preserved. Moreover, embodiments of theinvention provide modeling techniques that account for geometricaluncertainties in modeling data. In one embodiment of the invention, aplurality of equiprobable coherent models may be generated, each modelcorresponding to a different one of a plurality of equal probablesolutions to uncertain data.

The geometry of a subsurface may be represented by an implicit functionfor example t(x,y,z) representing the geological time of deposition ofthe particle of sediment which corresponds to the current geologicallocation (x,y,z). Horizons typically correspond to level surfaces oft(x,y,z) and faults typically correspond to the discontinuities oft(x,y,z). According to some embodiments of the invention, the followingmay be achieved:

-   -   optimal scaling of the geological-time function;    -   modeling geological properties, for example, the style of faults        (e.g., normal or reverse faults);    -   Removing erroneous or “pathological” data from geological models        such as local maxima and minima values of the geological-time        function on level surfaces, which correspond to non-realistic        geological horizons;    -   Generating a plurality of equiprobable coherent geometrical        models, which may be derived by coherent stochastic        perturbations of a preferred geometrical model; and    -   Displaying the results of stochastic perturbations.

In an embodiment of the invention, a computing system or one or moreprocessor(s) may perturb an initial three-dimensional (3D) geologicalmodel using a 3D vector field. The for example coherent 3D vector fieldincluding 3D vectors may be generated where each 3D vector of the 3Dvector field may be associated with a node of the initial 3D geologicalmodel and has a magnitude within a range of uncertainty of the node ofthe initial 3D geological model associated therewith. The coherent 3Dvector field may be applied to the initial 3D geological modelassociated therewith to generate a perturbed or alternative 3D model.The perturbed 3D model may differ from the initial 3D geological modelby a displacement defined by the 3D vector field associated with nodeshaving uncertain values. The perturbed 3D model may be displayed,stored, transmitted, etc.

In an embodiment of the invention, the computing system or processor(s)may apply separately one or more additional different 3D vector fieldsto the initial 3D geological model to generate a plurality of perturbed3D models. The processor may generate the 3D vector field to be coherentsuch that the divergence of the 3D vector field is greater than (−1).The processor may generate the 3D vector field to be coherent such thatfor any infinitely small polyhedral volume having vertices and faces inthe initial 3D geological model, the vertices are never displaced acrossany of the polyhedron faces. The processor may generate the 3D vectorfield to be equal to the gradient of a 3D scalar field. The processormay represent the 3D vector field at the nodes of a first mesh andrepresent the initial 3D geological model by a second mesh, wherein theprocessor displaces the nodes of a second mesh by applying the 3D vectorfield to the initial 3D geological model. The processor may generate theinitial 3D geological model to have three dimensions and to generate the3D vector field associated with each node of the 3D geological model tobe defined by three values, each of which corresponds to a different oneof the three dimensions of the 3D geological model. In some embodiments,the uncertainty at the nodes of the initial 3D geological modelcorresponds to uncertainty of a location of a fault. In otherembodiments, the uncertainty at the nodes of the initial 3D geologicalmodel corresponds to uncertainty of a seismic velocity field. Theuncertainty at the nodes may be proportional to the wavelength of datareflected from a subsurface geological feature. A receiving unit mayreceive a set of data reflected from a subsurface geological feature,wherein the initial 3D geological model uses the reflected set of datato model the subsurface geological feature. The processor may generatethe vector field to be a combination of two or more distinct vectorfields, each of which is associated with a different type ofuncertainty. The processor may define each of the nodes of the initial3D geological model to hold or be associated with a first parameterproportional to geological time and may perturb a set of the nodes byassociating or adding a second parameter to each node, wherein thederivative of the second parameter relative to the first parameter isgreater than (−1).

In an embodiment of the invention, a computing system or processor(s)may identify erroneous points in a geological model. A set of points ofthe geological model may be determined at which a geological-timefunction has a maximum or minimum value in a set of local regions. Themodel may be displayed on a user interface. The model may includesymbols indicating the location of each point in the set of points.Input may be accepted from a user. The set of points may be refinedaccording to input received from a user by adding or removing points inthe set of points. A model generated using the refined set of points maybe displayed to the user.

In an embodiment of the invention, a computing system may refine a firstgeological model. A first set of nodes may be identified at which ageological-time function has a local maximum or minimum value in thefirst geological model. A second set of nodes may be generatedcorresponding to the first set of nodes. The values of thegeological-time function at the second set of nodes may be differentthan the values of the geological-time function at the first set ofnodes. In the first geological model, the values at the first set ofnodes may be replaced with the values at the second set of nodes. Thefirst geological model in which the values of the first set of nodes arereplaced with the values at the second set of nodes may be displayed.

In an embodiment of the invention, a computing system may generate amodel by defining a combination of geological properties of the model.For each node of the model a value may be defined for each of aplurality of distinct geological properties associated with the modelincluding, for example, the geological time. For each location of themodel a linear combination of the values of the plurality of geologicalproperties at neighboring nodes may be generated. The model may bedisplayed.

In an embodiment of the invention, a computing system, in a 3Dsubsurface model, may rescale a function of time of when a geologicalstructure was originally formed in the Earth. A monotonically increasingfunction may be generated. A rescaled geological time function may begenerated by applying a rescaling function to the monotonicallyincreasing function. A 3D model of the structure may be displayed whenit was originally formed. The 3D model may be defined by the rescaledgeological time function in one of the three dimensions of the model.

In an embodiment of the invention, a computing system may model aprobability of leakage across a fault in a model. For each fault nodeassociated with the modeled fault, two or more nodes may be identifiedthat are substantially adjacent to the fault node and that are locatedon opposite sides of the fault from each other. A region of the modelrepresenting a subsurface structure having hydrocarbons may beidentified. The probability of leakage across the modeled fault may bedetermined for each fault node based on whether or not the two or morenodes are determined to be located in the identified region of themodel. The probability of leakage associated with each fault node may bestored.

BRIEF DESCRIPTION OF THE DRAWINGS

The principles and operation of the system, apparatus, and methodaccording to embodiments of the present invention may be betterunderstood with reference to the drawings, and the followingdescription, it being understood that these drawings are given forillustrative purposes only and are not meant to be limiting.

FIG. 1 is a schematic illustration of a uvt-transformation between two3D spaces;

FIG. 2 is a schematic illustration of a plurality of horizons;

FIGS. 3A and 3B are schematic illustrations of normal fault and areverse fault, respectively;

FIG. 4 is a schematic illustration of sampling points along well pathsin a model according to an embodiment of the invention;

FIG. 5 is a schematic illustration of the geometry of a horizonaccording to an embodiment of the invention;

FIGS. 6A and 6B are schematic illustrations of exploded and assembledperspective views of a mesh, respectively, according to an embodiment ofthe invention;

FIGS. 6C and 6D are alternate views of the illustrations of FIGS. 6A and6B, respectively;

FIG. 7 is a schematic illustration of a graphical user interface foraccepting user input according to an embodiment of the invention;

FIG. 8 is a schematic illustration of a display of a plurality ofequiprobable perturbed models according to an embodiment of theinvention;

FIGS. 9A and 9B are schematic illustrations of faults across whichhydrocarbon deposits may not and may leak, respectively, from one sideof the fault to the other side of the fault according to an embodimentof the invention;

FIG. 10 is a schematic illustration of a system according to anembodiment of the invention; and

FIGS. 11-16 are flowcharts of methods according to embodiments of theinvention.

For simplicity and clarity of illustration, elements shown in thedrawings have not necessarily been drawn to scale. For example, thedimensions of some of the elements may be exaggerated relative to otherelements for clarity. Further, where considered appropriate, referencenumerals may be repeated among the drawings to indicate corresponding oranalogous elements throughout the serial views.

DETAILED DESCRIPTION

For the sake of clarity and for the purpose of simplifying thepresentation of embodiments of the invention, the following preliminarydefinitions are given, although other definitions may be used:

Geological-Time

A particle of sediment in a geological terrain may be observed at alocation in the subsurface. The location of the particle may bemathematically represented or modeled, e.g., by a vector, (x,y,z), in athree-dimensional (3D) space, such as the Cartesian coordinate system(of course, when modelling such particles, the position of manyparticles may be modeled together as for example a cell). When modeled,a data structure such as a node or cell may represent particles. Thetime when the particle of sediment was originally deposited may bereferred to as the “geological-time” and may be represented or modeled,e.g., as a geological-time function of the current location of theparticle, t(x,y,z). When used herein, a “current” location for aparticle (or data structure representing one or more particles) orsubsurface feature may mean the location of the item in the present day,relative to geological time. The actual geological-time of thedeposition of particles may be difficult to determine and may bereplaced, e.g., by any arbitrary monotonic increasing function of theactual geological-time. It is a convention to use an arbitraryincreasing function, but similarly an arbitrary monotonic decreasingfunction may be used. The monotonic function may be referred to as the“pseudo-geological-time”. Geological-time and pseudo-geological-time arereferred to interchangeably herein.

Horizon, Fault and Unconformity

In stratified layers, horizons, faults and unconformities may becurvilinear surfaces which may be for example characterized as follows.

-   -   A horizon, H(t), may be a surface corresponding to a plurality        of particles of sediment which were deposited approximately at        substantially the same geological-time, t.    -   A fault may be a surface of discontinuity of the horizons that        may have been induced by a relative displacement of terrains on        both sides of such surfaces. In other words, the        geological-time (t) of deposition of the sediments is        discontinuous across each fault. Faults may cut horizons and may        also cut other faults.    -   An unconformity may be a surface of discontinuity of the        horizons that may have been induced by for example an erosion of        old terrains replaced by new ones. In other words, similarly to        faults, the geological-time (t) of deposition of the sediments        is discontinuous across each unconformity. When discussed        herein, unconformities are treated as faults: as a consequence,        in this patent application, faults may include both real faults        and unconformities. Alternatively, unconformities may be        surfaces bounding a sequence of sedimentary layers and one        specific geological-time function may be assigned to each such        sequence.        Reference Horizons and Control-Points

The geological domain may be characterized by a series of “referencehorizons” {H(t₁), H(t₂), . . . , H(t_(n))}, which may be separategeological layers with significantly distinct physical properties andassociated with distinct geological-times of deposition {t₁, t₂, . . . ,t_(n)}, respectively. When discussed herein, the horizons are implicitlyassumed to be sorted according to their geological-times of depositionso that H(t₁) is assumed to correspond to the older sediments andH(t_(n)) to the younger. In other words, the deposition time t_(i) ofH(t_(i)) is older than the deposition time t_(i+1) of H(t_(i+1)). Inpractice, each reference horizon H(t_(i)) is observed on a set ofsampling points referred to as “control-points” at which thegeological-time function may approximate t_(i) as closely as possible.Control-points may be a set of points generated from seismic data andwell data which have been identified, e.g., automatically by a computerand/or manually by a user, as representing to the same horizon, fault,or other geological structure.

Level Surface

The geological domain may be defined in a 3D space by a given functionof geological-time, t(x,y,z). The geological-time function t(x,y,z) maybe monotonic, i.e., the gradient of the geological-time never vanishesand the function has no local maximum or minimum values. A levelsurface, H(t₀), may be the set of points where the geological-timet(x,y,z) is equal to a given numerical value, t₀. Therefore, if thegeological-time t(x,y,z) represents a pseudo-geological-time ofdeposition, then the level surface H(t₀) of t(x,y,z) may be a geologicalhorizon.

Various mechanisms are currently used for modeling subsurface geologicalterrains:

Discrete-Smooth-Interpolation (DSI)

Discrete-Smooth-Interpolation (DSI) is a method for interpolating orapproximating values of a function f(x,y,z) at nodes of a 3D mesh, M,while honoring a given set of constraints. The DSI method allowsproperties of structures to be modeled by embedding data associatedtherewith in a (e.g., 3D Euclidean) modeled space. The function f(x,y,z)may be defined by values at the nodes of the mesh, M. The DSI methodallows the values of f(x,y,z) to be computed at the nodes of the mesh,M, so that a set of one or more (e.g., linear) constraints aresatisfied.

Constraints are typically classified by one of the following types:

-   -   1) “Soft” constraints may require the function f(x,y,z) to        approximate a constraint. For example, a soft constraint may be        that f(x,y,z) takes a given value at a given sampling point.        Since this constraint is “soft”, f(x,y,z) may take a least        squares approximation of the given value;    -   2) “Hard” constraints may require that the function f(x,y,z)        exactly honors the constraints. For example, a hard constraint        may include inequality constraints; and    -   3) “Smoothness” constraints may require that the function        f(x,y,z) varies as smoothly as possible while honoring the soft        and hard constraints.

Examples of DSI constraints include, for example:

-   -   The Control-Point constraint which may require that, at a given        location (x₀,y₀,z₀) the value of f(x₀,y₀,z₀) is equal to a given        value f₀;    -   The Control-Gradient constraint specifying that, at a given        location (x₀,y₀,z₀) the gradient of f(x₀,y₀,z₀), (∇fx₀, ∇fy₀,        ∇fz₀), is equal to a given vector G₀;    -   The Gradient-Direction constraint specifying that, at a given        location (x₀,y₀,z₀), the gradient of f(x₀,y₀,z₀), (∇fx₀, ∇fy₀,        ∇fz₀), is parallel to a given vector D₀; and    -   The Delta constraint specifying that, given two locations        (x₀,y₀,z₀) and (x₁,y₁,z₁) in the studied domain, the difference        between f(x₀,y₀,z₀) and f(x₁,y₁,z₁) may either be equal to,        lesser than, or greater than, a given value D.        GeoChron Model

When a layer of particles was deposited during a certain time period inthe past, the layer typically had uniform properties, such as particledensity, porosity, etc. However, through time, the layers erode and aredisrupted by faults, tectonic motion or other sub-surface movements,which result in uneven and discontinuous layers. As compared to theuniform layers of the past, the discontinuous layers of the present aredifficult to model. Accordingly, the “GeoChron” model has recently beendeveloped to operate between two 3D spaces. The 3D spaces or models maybe, for example:

-   -   A 3D space, G, representing a model of the current subsurface        features (e.g., the current modeled locations of particles of        sediment in the terrain). The modeled location of each particle        may be represented by the coordinates (x,y,z), where (x,y) may        describe the geographical coordinates of the particle (e.g.,        latitude and longitude) and (z) may describe the altitude or        distance below a surface level; and    -   A 3D space, G*, representing modeled locations of particles of        sediment at the time when the particles were originally        deposited. The modeled location of each particle may be        represented by the coordinates (u,v,t) where (t) may be the        geological-time of deposition of the particle and (u,v) may be        the paleo-geographical coordinates of the particle at geological        time (t).

The GeoChron model defines a transformation between the two 3D spaces.The transformation may be referred to as a “uvt-transformation”. TheGeoChron model applies the forward uvt-transformation to transform thecurrent model (a model of the subsurface features current in time) inG-space to the original deposition model in G*-space and applies theinverse uvt-transformation to transform the original deposition model inG*-space to the current model in G-space. Accordingly, the GeoChronmodel may execute complex computations on the original deposition modelin G*-space where geological properties are typically uniform and simpleto manipulate relative to the discontinuous current model in G-space.Once the original deposition model is sufficiently accurate, theGeoChron model may transform the model back to the current time domainto generate the current model.

Reference is made to FIG. 1, which schematically illustrates auvt-transformation between the two 3D spaces G and G* (separated bydashed line 101).

The “forward” or “direct” uvt-transformation 100 defines parametricfunctions 102 {u(x,y,z),v(x,y,z),t(x,y,z)}, which transform each point(x,y,z) of the geological G-space 104 to a point{u(x,y,z),v(x,y,z),t(x,y,z)} in the depositional G*-space 106. Theforward uvt-transformation 100 may be represented, for example, asfollows:

$\begin{matrix}{\left( {x,y,z} \right)\overset{UVT}{\longrightarrow}\left\{ {{u\left( {x,y,z} \right)},{v\left( {x,y,z} \right)},{t\left( {x,y,z} \right)}} \right\}} & \lbrack 1\rbrack\end{matrix}$

The forward uvt-transformation 100 transforms each horizon H(t) 108 aand 108 b of model 108, in the G-space 104, into a level horizontalplane H*(t) 110 a and 110 b of model 110, respectively, in the G*-space106. In the G*-space 106, horizons 110 a and 110 b of model 110 aresimply the images of level surfaces of the function t(x,y,z)representing the geological-time at location (x,y,z) in the G-space.That is, since a horizon models a set of particles of sediment that wasuniformly deposited in time, each horizon is constant in time when theparticles modeled thereby were originally deposited (i.e., in G*-space106). Therefore, each of horizons 110 a and 110 b in G*-space 106 may beuniquely defined by a single time, (t).

Conversely, the “inverse” or “reverse” uvt-transform 112 definesparametric functions 114 {x(u,v,t), y(u,v,t), z(u,v,t)}, which transformeach point (u,v,t) of the depositional G*-space 106 to a point{x(u,v,t), y(u,v,t), z(u,v,t)} in the geological G-space 104. Theinverse uvt-transformation 112 may be represented, for example, asfollows:

$\begin{matrix}{\left( {u,v,t} \right)\overset{{UVT}^{- 1}}{\rightarrow}\left\{ {{x\left( {u,v,t} \right)},{y\left( {u,v,t} \right)},{z\left( {u,v,t} \right)}} \right\}} & \lbrack 2\rbrack\end{matrix}$

In practice, such a reverse transformation is restricted to a part G*0of the G*-space called the holostrome and corresponding to the particlesof sediment which can actually be observed today in the G-space.

Using the forward uvt-transform 100, e.g., defined in equation (1), andthe inverse uvt-transform 112, e.g., defined in equation (2), anygeological property may be modeled in one of the two spaces (G-space 104or G*-space 106) and the result of the property modeled in the one spacemay be transferred to the other space (G*-space 106 or G-space 104,respectively). In practice, a geological property is typically modeledin the space where modeling the property is the simplest.

Modeling the Horizons

The following modeling process may generate the geological-time functiont(x,y,z) in the geological domain of interest (e.g., the domain beingmodeled, which may be generated automatically or may be selected by auser, as described herein). An approximation method, such as forexample, the DSI method or a radial approximation method may be used.

-   -   1) The geological domain of interest may be covered by a 3D mesh        M₁ including adjacent 3D polyhedral cells whose edges are        tangent to fault surfaces but do not cross these fault surfaces.        In a preferred embodiment, the cells may include tetrahedra or        hexahedra shaped cells, although other shapes may be used.    -   2) The geological-time function, t(x,y,z), is assumed to be        fully defined at the vertices (i.e., the nodes) of mesh M₁. The        values of the time function t(x,y,x) sampled at the vertices of        the cell containing the point (x,y,z) may be used to locally        (e.g., linearly) approximate the geological-time function,        t(x,y,x).    -   3) A “control-points” constraint may be used to require that,        for each sampling point (x,y,z) located on a reference horizon        H(t_(i)), the geological-time function t(x,y,z) will be        approximately equal to a constant value, t_(i).    -   4) Constraints may be used to require that the geological-time        function t(x,y,z) is as smooth as possible.    -   5) The approximation method may then be applied to compute the        values of t(x,y,z) at the vertices of mesh M₁ while honoring the        requirements of the aforementioned constraints. For example, the        DSI method or any other equivalent approximation method may be        used.

The geological-time function t(x,y,z) is initially defined only at thevertices of mesh M₁. The geological-time function t(x,y,z) for otherpoints (x,y,z), e.g., located inside a cell of mesh M₁, may be locallyapproximated from the defined values of t(x,y,z) at the vertices.

Although this process models the geological-time function t(x,y,z),problems may arise in the time function. Since each horizon represents aset of particles that were uniformly deposited in time (at the time ofdeposition), each horizon should be a level surface across which thegeological-time function is substantially constant. The geological-timefunction t(x,y,z) preferably has no local maximum or minimum values. Alocal maximum or minimum value at a point implies that the gradient ofthe geological-time function vanishes at these points, which in turnimplies that the geological-time stops at these points, a clearviolation of the forward moving principle of time. The aforementionedprocess of steps for modeling horizons may provide such a violation. Dueto outlier data and rapid lateral variations in the thickness ofgeological layers, the process may generate a geological-time functiont(x,y,z) having local maximum or minimum on horizons (i.e., levelsurfaces). Reference is made to FIG. 2, which schematically illustratesa plurality of horizons H(t_(i)) 200 generated by this process. Sinceeach horizon H(t_(i)) represents a layer of particles that wereuniformly deposited in time, in G-space each horizon should be a levelsurface across which the geological-time function t(x,y,z) should beconstant. However, the horizons H(t_(i)) 200 in FIG. 2 are not levelsurfaces. The horizons H(t_(i)) 200 include bumps 210 generated by thelocal maxima and minima in the geological-time function t(x,y,z), whichmay render this geological-time function incorrect.

Accordingly, there is a great need in the art for a mechanism to model avalid geological-time function, t(x,y,z), that does not include localmaximum or minimum in the studied domain.

GeoChron Model for Faults

To model discontinuities in a geological-time function t(x,y,z) acrossfaults a discontinuous mesh M₁ may be used. As shown in FIGS. 6A and 6B,a mesh 600 may be discontinuous, for example, in the following sense:

-   -   Edges of cells in the mesh M₁ may be tangential to fault        surfaces but may not cross these fault surfaces.    -   Collocated vertices of adjacent cells on both sides of a fault        may be duplicated in order to store distinct values of        geological properties such as, for example, t(x,y,z) at these        vertices for each side of the fault. Therefore, discontinuity in        geological properties across the fault may be modeled.

In a subsurface terrain, there are many types of faults, for example,branching faults, crossing faults, normal faults, reverse faults, etc.Current modeling mechanisms do not describe the structural behavior offaults. Accordingly, there is a great need in the art for a mechanism tomodel not only the location of faults, but the structural behavior offaults themselves. Accordingly, there is now provided with embodimentsof this invention an improved mechanism to model both the location offaults and the structural behavior of faults themselves (e.g., asdiscussed in the section entitled “Unified Modeling of a Geological-TimeFunction Integrating Multiple Sources of Information”) for effectivelyovercoming the aforementioned difficulties and longstanding problemsinherent in the art.

Geometrical Uncertainties and Preferred-Model

There are many sources of uncertainties in modeling geological data.When geological data is uncertain, a model computed using the data maybe uncertain. For models having an uncertain property (e.g., the seismicvelocity), a plurality of different models may be generated, eachalternative model having a different one of the possible solutions tothe uncertain property. Typically, each alternative model is generatedby applying different perturbations to an initial model. Alternativemodels may be generated according to principles of perturbation theoryand are thereby referred to as ‘perturbed’ models and the disturbanceapplied to the initial model are referred to as a ‘perturbation’.

Modeling Uncertainties on a Surface-Based Model

Common approaches to modeling the subsurface include building a set ofsurfaces corresponding to faults and reference horizons. For example,these surfaces may include faces or portions of adjacent polyhedracells. A model may be “perturbed” by, for example, moving each node (n)in the model, for example, by amount e(x_(n),y_(n),z_(n)|m) in directionD(n). To perturb such a geometric model, the following steps have beenproposed.

-   -   1) Select a random number (m) to be used as a “seed” by a random        function generator.    -   2) For each surface S of a model (where S may be a horizon or a        reference horizon):        -   a. Define a mesh M to be the set of nodes corresponding to            vertices of polygonal facets of S.        -   b. Use a stochastic simulator to generate an equiprobable            occurrence e(x_(n),y_(n),z_(n)|m) of a scalar field modeling            the magnitude of the perturbation at location            (x_(n),y_(n),z_(n)) of node (n) belonging to mesh M.        -   c. For each node (n) of M:            -   i. Choose a constant (non stochastic) direction of                perturbation D(n)            -   ii. Move the node in the direction of D(n) with a                magnitude of displacement equal to                e(x_(n),y_(n),z_(n)|m).    -   3) As far as a new stochastic occurrence of the geometric model        is required, return to step (1)    -   4) Stop

Models perturbed by this process are often geologically incorrect or“incoherent”. For example:

-   -   Faults may collide;    -   Horizons may cross each other; and    -   The connection between faults and horizons may have gaps.

From a structural geological perspective, such incorrect occurrencesrender the models difficult to use.

Modeling Uncertainties on Horizons of a GeoChron Model

Geological horizons having uncertainties may be modeled as levelsurfaces of a geological-time function t(x,y,z). For example, thegeological horizons may be modeled using the GeoChron model as discussedin the section entitled “Modeling the Horizons.” The following steps areproposed to stochastically model the uncertainties related to theposition of horizons:

-   -   1) Generate a model of the geological horizons using a        geological-time function t(x,y,z) defined at nodes of a 3D mesh        M₁.    -   2) Select a random number (m) to be used as a “seed” by a random        function generator.    -   3) Use a stochastic simulator to generate an equiprobable        occurrence e(x,y,z|m) of an error, i.e., the magnitude of the        perturbation for the occurrence, (m), affecting the        pseudo-geological-time at any node of the mesh M₁ in the studied        domain.    -   4) At each node of the mesh, compute the pseudo-geological-time        t(x,y,z|m) associated with the model (m) as follows:        t(x,y,z|m)=t(x,y,z)+e(x,y,z|m)    -   5) If a new equiprobable model of stochastic horizons is needed,        return to step (2) above. If no new equiprobable model of        stochastic horizons is needed, proceed to step (6).    -   6) Stop

Models perturbed by this process are often geologically incorrect. Forexample, the resulting geological-time function t(x,y,z|m) may havelocal minima or maxima across horizons. As a consequence, some“horizons” corresponding to level surfaces of t(x,y,z|m) may includeclosed surfaces around these local minima or maxima.

Modeling Uncertainties on Faults of a GeoChron Model

Faults having uncertainties may be modeled by generalizing theaforementioned process described in the section entitled, “ModelingUncertainties on a Surface-Based Model”, for example as follows.

-   -   1) For each fault F:        -   a. Assume that F includes of a set of polygonal (e.g.,            triangular) facets.        -   b. Define a set M_(F) to be a set of vertices of the            triangles of F.        -   c. Assume that each vertex (n) in M_(F) may be perturbed in            a given constant direction D(F).    -   2) Cover the studied domain with a 3D mesh M made of tetrahedral        cells, such that the mesh M contains M_(F).    -   3) Select a random number (m) to be used as a “seed” by a random        function generator.    -   4) Build a copy M(m) of the 3D mesh M.    -   5) For each fault F:        -   a. Define the direction of the stochastic displacement            v(n|m) of each vertex (n) of F in the direction of D(F).            Define the magnitude of the stochastic displacement v(n|m)            of each vertex (n) of F as a random spatially-correlated            scalar function e(n|m). For example, v(n|m)=e(n|m)·D(F),            where the symbol “(e·D)” represents the product of a            scalar (e) by a vector (D). For each tetrahedron cell            T(n₁,n₂,n₃,n₄) in M, apply a constraint to require, for any            displacement of nodes n₁, n₂, n₃ and n₄ by v(n₁|m)·D(F),            v(n₂|m)·D(F), v(n₃|m)·D(F), v(n₄|m)·D(F), respectively, that            the sign of the volume of tetrahedron cell t (i.e., (+) and            (−)) preferably does not change. Accordingly, the cells of a            tetrahedral mesh are not inverted.        -   b. Approximate v(n|m) on M in 3D for each n and m.        -   c. Move the nodes n of M according to the 3D approximation            of v(n|m). If a node n is located on both the fault F and on            another fault P, e.g., on which F branches, the direction            vector D(F) may be projected onto fault P before the nodes            are moved. This will prevent the nodes n from modifying the            general shape of P.    -   6) If a subsequent stochastic occurrence, (m), is needed, repeat        step (3) above to generate a new equiprobable model of        stochastic horizons. Otherwise, proceed to step (7).    -   7) Stop

Some steps are taken in this process to ensure the new equiprobablemodels are geologically correct. For example, as described above in step5(b), an approximation constraint may be used to ensure volumes oftetrahedral cells do not change sign. A post processing of theinterpolation result, as described above in step 5(d), may be used toensure that the displacement vector for fault branchings is projectedonto main faults. However, this process may still suffer from thefollowing drawbacks:

-   -   Only a mesh composed of tetrahedral cells, not a general mesh        with other types of cells, may be used because the process        requires a tetrahedral tessellation of the studied domain.    -   A single (constant) predefined direction of displacement D(f) is        used for each fault.    -   Since the direction of displacement is constant, projections of        displacement vectors at fault branchings are made after        approximation. This may result in abrupt changes in the shape of        branching faults near main faults.        Modeling Uncertainties on Horizons of a Potential Field Model

Another mechanism for modeling uncertainties in geological horizons usespotential fields. This mechanism has several drawbacks:

-   -   Initially, a “covariance” function defining the stochastic        behavior of the horizons may be estimated. However, there is        typically insufficient data collected in seismic surveys to        accurately estimate the covariance function.    -   When modeling a complex network of hundreds of faults, modeling        each fault may be impractical.    -   Taking into account the uncertainty of faults and fault        positioning may be difficult.    -   Ensuring that the horizons corresponding to level surfaces of        the potential field are open surfaces may be difficult. Due to        outlier data or due to rapid lateral variations of the thickness        of geological layers, some level surfaces of the potential field        may have local maxima or minima.

The aforementioned techniques allow the inherent uncertainty in seismicdata to be modeled. If a model is uncertain, a unique solution istypically indeterminate at one or more nodes of the model and at best, arange of possible solutions having an equal probability of being correctmay be determined at each node. These techniques propose perturbing themodel for each equal probability solution.

Each node may have a different level of uncertainty. The greater theuncertainty of a node, the greater the difference between equalprobability solutions. For example, if the location of a first faultdefined on a first node is highly uncertain, the possible locations ofthe first node may span a great distance. Conversely, if the location ofa second fault defined on a second node is highly certain, then thepossible locations of the second node will be bound to a small region.Accordingly, to generate a new model, each node, n, may be moved by adifferent distance, e(n|m), based on the level of uncertainty at thatnode.

A plurality of models having equal probability of occurrence isgenerated to view each possible solution to an uncertain problem, wheresuch information of the possible solutions would otherwise have beenlost. However, these mechanisms still discard significant data. Thesemechanisms move all nodes of a fault surface F in the same(non-stochastic) direction, D(F), i.e., in one dimension. However, sinceuncertainties, e.g., fault location or other parameters, occur in threedimensions and are typically not limited to one, moving all nodes of asurface in only one dimension may neglect the uncertainties of otherdimensions. Accordingly, information pertaining to alternative equalprobability solutions for uncertainties in other dimensions will neverbe displayed to a user.

Accordingly, there is a great need in the art for a mechanism to modelequal probable solutions for uncertainties in greater than onedimension, while preserving the geological coherency of these solutions.

For simplicity and clarity of illustration, embodiments of the inventionare described in two separate sections (elements from each may of coursebe combined in some embodiments).

-   -   The first section, entitled ‘Generating a Coherent Preferred        Model’, describes mechanisms which may be used according to        embodiments of the present invention to generate a structurally        coherent “preferred-model” of horizons and faults.    -   The second section, entitled ‘Modeling Uncertainty Related To        Geological-Time Function’, describes mechanisms which may be        used according to embodiments of the present invention to        generate a series of equiprobable geometrical models including        of coherent perturbations of the preferred-model.

Embodiments of methods for processing subsurface data are describedherein. Typically, input data to these methods includes, for example,seismic data, well data, acoustic data, or other data, which may be usedby processes described herein, and other known processes, to modelsubsurface geological features.

Generating a Coherent Preferred Model

Rescaling the Geological Time Function

Geological-time values associated with reference horizons are typicallychosen arbitrarily, for example, where geological-time values increaseas the assumed deposition time increases or alternatively, wheregeological-time values decrease as the assumed deposition timeincreases. However, choosing these reference values arbitrarily maycause numerical instabilities induced, e.g., by the presence of strongvariations of thickness between adjacent layers bounded by referencehorizons. Embodiments of the invention include rescaling an initialgeological-time function t(x,y,z) based on for example arbitraryreference values. For example, the rescaled geological-time functiont(x,y,z) may be defined so that the norm of the gradient of the rescaledgeological-time function t(x,y,z) remains, on average, approximatelyconstant in the studied domain.

The initial “guess” or geological-time function based on arbitraryreference values t₀(x,y,z) may be rescaled by a rescaling function F(t₀)to generate a rescaled geological-time function t=F(t₀), e.g., describedas follows.t(x,y,z)=F(t ₀(x,y,z))  [3]

The rescaling function F(t₀) may transform the initial geological-timefunction t₀(x,y,z) into a rescaled geological-time t(x,y,z). In oneembodiment of the invention, the norm of the gradient of the rescaledgeological-time function may be, on average, approximately equal to agiven positive constant, g. The constant, g, may be, for example, 1,although other values may be used.

The gradient of the rescaled geological time t(x,y,z) may be derived atany location (x,y,z) from the gradient of the initial geological timet₀(x,y,z), for example as follows:grad t(x,y,z)=F′(t ₀(x,y,z))·(grad t ₀(x,y,z))  [4]where F′ (t₀) represents the derivative of F(t₀) and “(a·v)” representsthe product of a scalar (a) by a vector (v).

If the norm of the gradient of the refined geological-time function,∥grad t(x,y,z)∥, is chosen to be equal to 1, on average, then equation[4] simplifies to the following equation:F′(t ₀(x,y,z))=1/∥grad t ₀(x,y,z)∥ on average  [5]For the sake of clarity, the following notations are used here:

t(i)=t_(i)

t₀(i)=t₀i

D(i)=the average of the ratio {1/∥grad t₀(x,y,z)∥} over the referencehorizon H(t₀i) Using this notation, equation [5] may be rewritten,approximately, as:F′(t ₀ i)=D(i)  [6]

Accordingly, an optimal rescaling function F(t₀) may be defined bysolving differential equation [6] by numerical integration. To solvedifferential equation [6] numerically, numerical techniques may be used,which are known in the art such as, for example, a Euler method or aRunge-Kutta method.

In one embodiment of the invention, the following steps may be used totransform initial geological times {t₀1, t₀2, . . . , t₀n} of theinitial arbitrary geological-time function t₀(x,y,z) of the referencehorizons into optimally rescaled times {t₁, t₂, . . . , t_(n)} of therescaled geological time t(x,y,z) (other operations or series ofoperations may be used in different embodiments).

1) initialize i=1;

2) initialize t(1)=arbitrary geological time (e.g., t(1)=0);

3) t(i+1)=t(i)+{t0(i+1)−t0(i)}*(D(i+1)+D(i))/2;

4) i=i+1;

5) if(i<=n) return to step (3), otherwise proceed to step (6); and

6) Stop.

Embodiments of the invention provide a system and method for improvingthe accuracy of a model, for visualizing geological data, by providing amore realistic geological-time function of the data. The geological-timefunction may be rescaled so that the norm of its gradient remains, onaverage, approximately constant or uniform throughout the model. Inputdata to the model may include seismic waves that after being transmittedby a transmitter (e.g., transmitter 190 in FIG. 10) are received by areceiving device (e.g., receiver 120 in FIG. 10). A processor may inputthe collected data to compute a geological-time function coincidingtherewith. However, the geological-time function may be inaccurate(e.g., the level set of the geological-time function which represents areference horizon may be greater than a predetermined distance from datapoints) at points computed using data collected, which were reflectingfor example from regions in the subsurface having large gradients inthickness between adjacent layers bounded by reference horizons. Tominimize such inaccuracies of the geological-time function, a rescalingfunction may be applied to an arbitrary monotonically increasinggeological-time function to generate a regularly increasing function oftime. The rescaled geological-time function may be stored (e.g., as data155 in memory 150 of FIG. 10) and used to generate models, as describedfor example, in the sections entitled, “Modeling Uncertainties onHorizons”, “Modeling Uncertainty Related to the Positioning of Particlesof Sediment”, and “Modeling Uncertainty on Faults and Horizons Jointly”.Models generated by the rescaled geological-time function may be moreaccurate than conventional models generated by an arbitrarygeological-time function, which is not optimally scaled. Accordingly, amore accurate visualization of the subsurface terrain may be presentedto a user (e.g., on display 180 of FIG. 10).

Unified Modeling of a Geological-Time Function Integrating MultipleSources of Information

The geometry of geological subsurfaces, defined by a geological-timefunction t(x,y,z), is both complex and poorly known, for example, as aresult of limited and sparse data provided by heterogeneous sources ofinformation. Due to a lack of generality, methods known in the arttypically use only part of all the available information to model thesubsurface, thereby neglecting other important sources of information.

In contrast to methods known in the art, embodiments of the inventionprovide a system and method for integrating multiple sources ofinformation to model a geological-time function in a unified way.

Sources of information frequently used for modeling geological-timefunctions, e.g., with level surfaces correspond to geological horizons,may include, for example, the following:

-   -   Sampling points: Each reference horizon H(t_(i)) may be defined        by a set SH(t_(i)) of sampling points {s₁, s₂, . . . } located        on horizon H(t_(i)). For example, these sampling points {s₁, s₂,        . . . } may be generated by methods known in the art such as by        “auto-picking”, in which the sampling points are automatically        extracted from a given seismic cube (or cell, node, vertex,        edge, column or other feature of the mesh). To use the sampling        point information, the model may specify that, for any sampling        point (s) in a set SH(t_(i)), the geological-time function        t(x,y,z) is equal to (t_(i)) at location (s). Sampling point        information is typically the most basic source of information        and is often the only source of information taken into account        by conventional methods of modeling the geological-time        function. According to embodiments of the invention, sampling        point information is just one of the multiple sources of        information that may be used to model the geological-time        function.    -   Geometry of Faults: In a subsurface terrain, there are different        types of faults, for example, normal faults and reverse faults,        etc. Conventional modeling mechanisms typically model spatial        characteristics of the subsurface terrain, but do not model the        structural behavior of faults. Embodiments of the invention        provide a mechanism to model not only the location of faults,        but also the structural behavior of faults themselves.    -   The geological-time function is typically discontinuous across        faults. Such discontinuities may be located and faults may be        spatially modeled. However, faults are complex structures with        many properties. Simply locating the faults in a subsurface may        neglect the geological behavior of the faults. For example,        fault surfaces are rarely exactly vertical and, depending on the        relative location of an arbitrary horizon H(t₀) on both sides of        a fault, the fault may be a normal fault or a reverse fault.        Reference is made to FIGS. 3A and 3B, which schematically        illustrate a normal fault 300 and a reverse fault 302,        respectively. Consider the vector N_(F) 304 which is, on        average, orthogonal to a fault F 300 or 302 and such that its        vertical component is positive. Such a vector N_(F) 304 allows        the positive and negative sides of fault F 300 or 302 to be        defined as follows: across the surface of faults F 300 and 302        in the direction of vector N_(F) 304, the side of fault 300 or        302 changes from negative to positive. Two points (n) and (p)        are collocated at any arbitrary fault point 306 along fault F        300 or 302 and on the negative and positive side of F,        respectively. The geological-time function of the points, t(n)        and t(p), may be defined. The geological behavior of the faults        300 or 302 may be determined as follows:        -   If t(p) is greater than t(n), then F is a “normal” fault            300;        -   If t(p) is less than t(n), then F is a “reverse” fault 302.    -   Normal faults 300 and reverse faults 302 may be alternatively        defined, for example, replacing collocated cells for collocated        points 306 p and n in the definition above.    -   Geometry of Well Paths: Reference is made to FIG. 4, which        schematically illustrates sampling points 402 a, 402 b, and 402        c in a model 400. As shown in FIG. 4, any sampling point 402 a,        402 b, or 402 c on a well path 404 may be classified, for        example, into one of the following categories, depending on the        location of the sampling point 402 a, 402 b, and 402 c relative        to a reference horizon H(t_(i)) 406 corresponding to a given        geological-time (t_(i)), for example, as follows:        -   a set WH(t_(i)) of sampling points {w₁, w₂, . . . } 402 a            located at intersections of the well path 404 with horizon            H(t_(i)) 406, i.e., well markers,        -   a set AH(t_(i)) of sampling points {a₁, a₂, . . . } 402 b            located above the horizon H(t_(i)) 406, and        -   a set BH(t_(i)) of sampling points {b₁, b₂, . . . } 402 c            located below the horizon H(t_(i)) 406.    -   In practice, such a classification of sampling points 402 a, 402        b, and 402 c is typically determined by geologists based on        observations of the nature of the terrains along well paths        although other automated mechanisms may also be used.        Conventional modeling tools typically only use sampling points        corresponding to set WH(t_(i)) 402 a that coincide with well        markers and not sampling points 402 b and 402 c corresponding to        sets AH(t_(i)) and BH(t_(i)), located above and below the well        path 404, respectively. Sampling points 402 b of sets AH(t_(i))        and 402 c BH(t_(i)) located above and below the well path 404        may be used to define the location of horizon H(t_(i)) 406. One        current method merges sets AH(t_(i)) with WH(t_(i)), but        neglects using information related to the set BH(t_(i)) of        sampling points below the horizon. Without the information        related to the set BH(t_(i)) of sampling points below the        horizon, the modeling mechanism typically generates a horizon        with an inaccurate (e.g., “bumpy”) surface. The inaccurate        surface of the horizon H(t_(i)) may require the modeling        mechanism to execute an additional processing step to correct        and smooth the surface of horizon H(t_(i)). For example,        heuristic iterative post-processing steps may be executed, which        may strongly depend on assumptions which are difficult to honor        in practice. According to an embodiment of the invention,        information relating to the geometry of well paths may be used,        for example, in connection with other sources of information, to        model the geological-time function. In one embodiment of the        invention, the information relating to the geometry of well        paths 404 may include information from the sets WH(t_(i)) 402 a,        AH(t_(i)) 402 b and BH(t_(i)) 402 c of sampling points that        coincide with, are located above, and are located below the        horizon H(t_(i)) 406, respectively.    -   Geometry of Horizons: Reference is made to FIG. 5, which        schematically illustrates the geometry of a horizon 506 of a        model 500, e.g., defined by the orientation of normal vectors        504 of the horizon 506 at sampling points 502 thereof. A normal        vector 504, N(s), which is orthogonal to the horizon 506 and        coincides with a sampling point (s) 502, may be defined for each        sampling point 502 in a set of sampling points S_(N)={s₁, s₂, .        . . }. In one embodiment of the invention, the sampling        points (s) 502 and the corresponding normal vectors N(s) 504 may        be selected by the geologist or user, for example, who may view        the model 500 on a graphical user interface (e.g., displayed on        a monitor or display) and select sampling points (s) 502 and the        normal vectors N(s) 504 by entering model coordinates or        selecting one or more points or directions using a keyboard,        mouse or other input device. In another embodiment of the        invention, the sampling points (s) 502 and the normal vectors        N(s) 504 thereof may be extracted automatically by a computing        device (e.g., from a given seismic cube). In still another        embodiment of the invention, the normal vectors N(s) 504 may be        measured at sampling location (s) 502 along a well path, as is        known in the art.    -   One current method describes using normal vectors N(s) for        modeling by equating the gradient of the geological-time        function with normal vectors N(s) at the locations of sampling        points (s). This requires the module ∥N(s)∥ and the orientation        of normal vectors N(s) to be known. However, the module ∥N(s)∥        is typically unknown and, according to one current method,        choosing an arbitrary value for the module ∥N(s)∥ may jeopardize        the approach. Furthermore, the orientation of normal vectors        N(s) is often uncertain and using an incorrect orientation for a        normal vector N(s) at just a single sampling point (s) of the        set of sampling points S_(N) is sufficient to trigger        devastating effects, thereby generating an inaccurate        geological-time function. According to an embodiment of the        invention, the gradient of the geological-time function may be        parallel to a given normal vector N(s) 504 at any sampling        point (s) 502 in the set of sampling points S_(N). In this        embodiment, neither the orientation of normal vectors N(s) 504        nor the corresponding module ∥N(s)∥ are used and may be chosen        arbitrarily for each sampling point 504 of S_(N). For example,        contrary to one current method, FIG. 5 shows that the normal        vector N(s₃) 504 at one sampling point 502 may be oriented in a        different direction relative to normal vectors N(s₁), N(s₂) and        N(s₄) at each different sampling point 502 (s₁), (s₂,) and        (s₄,), respectively.    -   Other possible sources of information may be used to model a        geological-time function.

For example, information may be used relating to seismic travel times,i.e., the time in which seismic rays travel to reach a given horizonH(t_(i)). Seismic travel times may be calculated using both the geometryof the rays and the seismic velocity field.

Embodiments of the invention include modeling a geological-time functiont(x,y,z) using one or more of the following sources of information, asdescribed above, and possibly other information:

-   -   Sets of sampling points {SH(t₁), SH(t₂), . . . } on reference        horizons {H(t₁), H(t₂), . . . };    -   Fault types (e.g., reverse faults 302 or normal faults 300);    -   Sets {WH(t₁), WH(t₂), . . . }, {AH(t₁), AH(t₂), . . . } and        {BH(t₁), BH(t₂), . . . } corresponding to sampling points 402 a,        402 b, and 402 c, respectively, along well paths 404, which        coincide with, are located above, and are located below the        horizon H(t_(i)) 406, respectively;    -   A set of sampling points S_(N) such that, for any sampling        point (s) 502 in S_(N) a normal vector N(s) 504 orthogonal to        the horizons 506 is given. By definition, N(s) is called the        “normal vector” at location (s); and    -   Other information, such as seismic travel times, which may be        translated into linear combinations of the values of the        geological-time function at the nodes of a given mesh.

Embodiments of the invention provide a system and method for integratingmultiple independent sources of information and simultaneously modelinga geological-time function t(x,y,z) using each of the informationsources together. A plurality of distinct geological properties, e.g.,corresponding to each of the multiple different information sources, mayhave value(s) at given sampling locations. Each of the plurality ofdistinct geological properties may be defined by a linear function orrelationship, such as equality or inequality. The plurality ofgeological properties may be combined by defining a linear system of thelinear functions for all or some of the distinct geological properties.The solution of such a linear system may be defined by the values of thelinear functions at each node of the mesh M₁. The geological-timefunction may be generated using one or more of the linear equations ateach node. Since the plurality of distinct geological properties arecombined into a linear system, all of the multiple independent datacorresponding to sources of information may be applied simultaneously,in a single execution of the modeling procedure. The geological-timefunction t(x,y,z) is thereby restricted at the same time to meet all ofthe multiple geological properties.

The geological-time function may be generated or determined using acombination of geological constraints being simultaneously applied, forexample, as follows (other operations or series of operations may beused, and the exact set of steps shown below may be varied).

-   1) The geological domain (e.g., that covers the physical space from    which seismic data is collected) may be covered by a corresponding    mesh M₁. Reference is made to FIGS. 6A and 6B, which schematically    illustrate exploded and assembled perspective views of a mesh 600,    respectively. Mesh 600 includes a plurality of cells 602. In the    views of FIGS. 6A and 6B, cells 602 are tetrahedral cells, although    other types of polyhedral cells such as hexahedral cells may be    used. Mesh 600 may be generated, for example, as is known in the    art.-   2) The geological-time function t(x,y,z) may be fully defined by its    values at the vertices of mesh M₁. The geological-time function may    be generated at an arbitrary point (x,y,z) by locally approximating    known values of the geological-time function sampled at the nodes of    M₁ neighboring the point (x,y,z). Moreover, the mesh M₁ may be    generated so that discontinuities of the geological-time function    t(x,y,z) across the faults are permitted (e.g., as shown in FIG. 6).-   3) For each reference horizon H(t_(i)) corresponding to particles of    sediment deposited at a given geological-time (t_(i)):    -   a) For each sampling point location (s) of the set SH(t_(i)),        apply a “control-points” constraint, which may require the        geological-time function t(x,y,z) to be approximately equal to        (t_(i)) at location (s).    -   b) For each sampling point location (w) of the set WH(t_(i)),        apply a “control-points” constraint, which may require the        geological-time function t(x,y,z) to be approximately equal to        (t_(i)) at location (w).    -   c) For each sampling point location (a) of the set WA(t_(i)),        apply an inequality constraint, which may require the        geological-time function t(x,y,z) to be greater than (t_(i)) at        location (a).    -   d) For each sampling point location (b) of the set WB(t_(i)),        apply an inequality constraint, which may require the        geological-time function t(x,y,z) to be less than (t_(i)) at        location (b).    -   The control-point constraints of steps (3a) and (3b) and the        inequality constraints of (3 c) and (3d) are typically linear        and, as a consequence, may be used together as a linear system        of constraints to model the geological-time function.-   4) For each sampling point location (s) in the set S_(N), an    additional or alternative constraint may be applied to model the    geological-time function, specifying that, at location (s), the    gradient of the geological-time function may be approximately    parallel to the given vector N(s). That is, the cross product of the    gradient of the geological-time function with N(s) may substantially    vanish (i.e., go to zero). Such a constraint may be represented as a    linear function and, as a consequence, may be used together in a    linear system with other constraints for modeling the    geological-time function.-   5) For each fault F whose type (normal or reverse) is given:    -   a) sample F with a set of pairs of points (n,p) collocated on        the negative and positive sides of F, respectively.    -   b) for each pair of sampling points (n,p), add one of the two        following inequality constraints:        -   i) If the fault F is a normal fault, then apply an            inequality constraint specifying that geological-time at            location (p) should be greater than geological-time at            location (n); and        -   ii) If the fault F is a reverse fault, then apply an            inequality constraint specifying that geological-time at            location (p) should be lower than geological-time at            location (n).    -   Other constraints or different sets of constraints may be used.        The inequality constraints of steps (5bi) and (5bii) are        typically linear functions and, as a consequence, may be used        together in a linear system with other constraints for modeling        the geological-time function.-   6) Other source of information may be represented by a linear    function and translated into linear combinations of the values of    the geological-time function at the nodes of M₁ for use as a    constraint together in a linear system with other constraints for    modeling the geological-time function.-   7) Additional constraints may be applied to specify that, e.g., over    the entire studied domain, the geological-time function t(x,y,z)    should vary as smoothly as possible between each node or cell of M₁    and any neighboring node or cell when there is no fault    therebetween.-   8) The DSI method or any other equivalent approximation method may    then be applied to compute the values of the geological-time    function t(x,y,z) at each node of mesh M₁ as solution of the linear    system of equations and inequalities induced by the constraints.    Since the value of the geological-time function t(x,y,z) at each    node is derived from linear equations and inequalities at each node,    the geological-time function is modeled based on the constraints    from multiple sources of information which define the linear    equations and inequalities of the system.-   9) Stop

In step (7) above, a “smoothness constraint” may be used, defined, forexample, by one or more of the following (other constraints may beused):

-   -   Smoothness constraint #1: In the neighborhood of each cell of        mesh M₁, the gradient of t(x,y,z) is constrained to be as        constant as possible;    -   Smoothness constraint #2: In the neighborhood of each node of        mesh M₁, the function t(x,y,z) is constrained to vary linearly,        as much as possible;    -   Smoothness constraint #3: In the neighborhood of each node of        mesh M₁, the function t(x,y,z) is constrained to be as close as        possible to a harmonic function.

Each of these smoothness constraints is preferably linear so that theconstraint may be easily implemented in the DSI method or any otherequivalent approximation method to further constrain the values of thegeological-time function t(x,y,z) at each node of mesh M₁.Alternatively, the constraints may be non-linear and may be approximatedas a linear function or used with an algorithm enabling non-linearconstraints.

A plurality of geological constraints may be selected from the groupconsisting of the occurrence of sampling points, the geometry of faults,the geometry of well paths, the geometry of horizons, and/or seismictravel times by executing the operations of corresponding steps 3-6,respectively (other constraints may be used). If any of these geologicalconstraints is undesirable, the corresponding step in the aforementionedprocess may be skipped. In one embodiment, a user may select which ofthe geological constraints are used. In one embodiment, there may be adefault constraint to generate a model and additional constraints may beoptionally selected by the user.

Reference is made to FIG. 7, which schematically illustrates a graphicaluser interface 700 for accepting user input according to an embodimentof the invention. A user may operate an input device (e.g., input device165 of FIG. 10), such a mouse or keyboard. The user may select a subsetof the sampling data representing a horizon to be decimated or removedfrom the set of sampling data F_(sp) if, for example, the sampling datais redundant, erroneous, or otherwise undesirable.

The aforementioned process may be executed entirely automatically and/orbased on user input, according to software instructions (e.g., insoftware 160 of FIG. 10) executed by a processor (e.g., processor 140 ofFIG. 10). Typically, the process is initiated automatically when theprocessor receives data generated by an intermediate process.Alternatively, the process initiates by a command via an input device(e.g., controlled by a user). Alternatively or additionally, the usermay control or modify the otherwise automated process or providespecific data replacing the data automatically generated or adding datathereto. The data provided by a user may relate to the structure offaults (e.g., normal or reverse) and/or other geophysical properties.Data related to geophysical properties of the model may be entered by auser moving pointing device (e.g., a mouse) to cause a computing systemto manipulate a visualization of the model (e.g., on display 180 of FIG.10). Alternatively, the user may enter values associated withgeophysical and geological properties at nodes or regions of the modelby entering such values in fields of a table, list, or other datastructure. The geophysical and geological properties entered by a usermay be automatically integrated by a processor as described herein togenerate a model based on the data entered by the user.

Characterizing “Bubbles”

The studied geological domain may be defined by a monotonicgeological-time function, t(x,y,z). A “bubble” may be a point of errorwhere the geological-time function t(x,y,z) has a local minimum ormaximum value on a horizon in the studied domain. A horizon is a surfacecomposed of a set of particles of sediment that were uniformly depositedat the same geological-time. At the time of original deposition, eachhorizon (e.g., horizons 110 a and 110 b in uvt-space of FIG. 1) shouldbe a level surface having a constant geological-time function. If thegeological-time function has a local minimum or maximum, then a “bubble”or closed level surface (e.g., a ring, a torus, a multiple loopedsurface) may exist around that local minimum or maximum. From ageological perspective, a bubble or closed surfaces cannot representvalid horizons. Therefore, the geological time function should not haveany points of maximum or minimum value on a horizon. However,conventional systems are known to generate these erroneous points.

Embodiments of the invention include one or more of detecting,displaying, and/or removing local minimum or maximum values of thegeological-time function t(x,y,z)) on horizon surfaces in the studieddomain.

Detecting “Bubbles”

To detect bubbles, e.g., points at which the geological-time functiont(x,y,z) has a local minimum or maximum value on a horizon in thestudied domain, the following steps may be executed (other steps orseries of steps may be used, and the specific set of steps below may bemodified in various embodiments).

-   -   1) Generate an initial geological-time function t(x,y,z) at the        nodes of a given mesh M₁, for example as described herein.    -   2) Generate (e.g., determine) a set P of erroneous or        “pathological” nodes of the mesh M₁ (e.g., nodes at which the        geological-time function t(x,y,z) has a local minimum or maximum        value). For each node (n) of mesh M₁, e.g., the following        example process may be iteratively used to determine if t(x,y,z)        has a local minimum or maximum value at the node (other        processes may be used):        -   a. Determine the coordinates (x_(n),y_(n),z_(n)) of the node            (n);        -   b. Compute the maximum value, tmax(n), and the minimum            value, tmin(n), of the geological-time function at any node            of the mesh M₁ directly connected to node (n) by, e.g., any            edge of the mesh M₁;        -   c. If {t(x_(n),y_(n),z_(n))>tmax(n)} or            {t(x_(n),y_(n),z_(n))<tmin(n)}, then add (n) to the set P of            pathological nodes.    -   3) Stop.

Embodiments of the invention include identifying erroneous data such asa maximum value, tmax(n), and a minimum value, tmin(n), of thegeological-time function in a neighborhood of a node (n). A neighborhoodof node (n), may for example, be a local region in which e.g., any nodedirectly connected to the node (n) by any edge, face, cell, or apredetermined number thereof defining a region of the mesh M₁.Alternatively, the neighborhood of node (n) may be a sphere of apredetermined radius, a region spanning predetermined coordinates, orany subspace of the modeled domain. The value of the geological-timefunction at each node is then compared to the local maximum and minimumvalues in the local area. If the value of the geological-time functionat a node is within a range (tmin(n), tmax(n)) bounded by the localmaximum and minimum values, then the node is properly defined andremains. However, if the geological-time value of a node is itself thelocal maximum or minimum, i.e., greater than the local maximum or lessthan the local minimum (e.g., and optionally if the node is equal to)the values at its neighboring nodes (excluding itself), then the nodemay be detected as such and added to a set of P of “pathological” nodesof the mesh M₁. These nodes may be displayed to investigate and/orremoved to overcome the aforementioned problems associated therewith, asfollows.

Visualizing or Displaying “Bubbles”

Due to for example erroneous data, rapid lateral variations of thethickness of geological layers, or other causes, a modeling process maygenerate bubbles. It may be important to visualize or display, andpossibly to edit these bubbles.

To display and visualize bubbles, e.g., on a graphical user interface,the following steps may be executed (other operations or series ofoperations may be used, and the exact set of steps shown below may bevaried).

-   -   1) Select the set P of erroneous or “pathological” nodes of the        mesh M₁, e.g., where there are “bubbles.”    -   2) Using any appropriate graphical user interface, e.g., of a        computing device, display the set of nodes belonging to P with a        predefined symbol, color, flag, or marker, etc. For example,        each pathological node may be represented as a small red sphere.    -   3) For each node in P, focus the display on that node, or        highlight that node, and offer the user the two following        options (other options may be offered):        -   a) If the user identifies (e.g., by inputting a selection            into a computer) an erroneous sampled data point in the            neighborhood of the pathological node, then the user may            remove that sampled data from the data set used to model the            geological-time function;    -   b) If the user determines that additional nodes may be        pathological, e.g., in a region of rapid variation of thickness        in the neighborhood of the pathological node, then the user may        refine the mesh M₁ in that neighborhood to better model the        variations of the geological-time function.    -   4) Stop

The symbols or markers used to indicate that nodes are local maxima orminima of the geological-time function may be overlaid on a spatialmodel of the corresponding nodes. In one embodiment, a user may selectto have the indicating symbols turned on or off. The symbols may beshown by any color or shape (e.g., a red sphere or cube), or by anyother deviation or change from the normal way the node is displayed(e.g., highlighting, size, etc.). In an alternate embodiment, multiplecolors and symbols or changes may be used to indicate multiplecategories of pathological nodes separated in different sets P and P′.For example, different sets P and P′ may include the local maxima orminima nodes for regions of the mesh having different sizes, e.g., reddots may indicate the local maxima/minima nodes in P of a relativelysmall region (e.g., nodes sharing vertices) and blue dots may indicatethe global maxima/minima nodes in P′ on each horizon.

A user may refine or perfect the process by which bubbles areautomatically detected, as described herein. A user may inspect thedisplay and determine that a node should be added or removed from theset of pathological nodes in the set P. For example, if there is a greatlikelihood that local maxima or minima may occur in a specific region(e.g., due to a rapid variation in the thickness of layers), the usermay highlight (e.g., by indicating with a pointing device or byotherwise entering data to a computer) or otherwise indicate thatadditional nodes may be extracted from the region. In one example, thehighlighted region may be refined or divided into smaller sub-regions,each of which may have its own pathological nodes. A computing modulemay send a request for input from the user or provide a user input fieldin a graphical user interface, e.g., as shown in FIG. 7. The user mayprovide input data such as a command received by computer module toexecute the aforementioned operations to detect pathological nodes, butthis time for the selected or smaller sub-regions. Additional nodesdetected to be local maxima or minima in each of the sub-regions mayoptionally be inspected for confirmation from the user and/or includedin the set P of pathological nodes. In another embodiment, the user mayhighlight a problematic region (e.g., having many pathological nodes)and a processor may detect and remove pathological nodes in thehighlighted region only, thereby restricting the model correctionprocess to correct nodes in an isolated region and minimally varying theremainder of the model. For example, the user may highlight and combinetwo or more regions into a single combined region and execute thepathological node detection for the larger region. This process mayeliminate pathological nodes selected for each region and only select orhighlight a single pathological node for all regions combined, which hasthe relatively largest maximum or minimum value in the combined region.

Once a user believes the proper representation of nodes is displayed fora particular model, the user may proceed to remove the pathologicalnodes, for example as follows.

Removing “Bubbles” Automatically

Nodes at which the geological-time function has a maximum or minimumvalue in a local region are known to be erroneous. These pathologicalnodes or “bubbles” may be removed from a model or mesh, for example,according to the steps that follow (other operations or series ofoperations may be used, and the exact set of steps shown below may bevaried).

-   -   1) Generate an initial geological-time function t₀(x,y,z) at the        nodes of a given mesh M₁, as described herein.    -   2) Select the set P of “pathological” nodes p of the mesh M₁        generated above.    -   3) Optionally, add or remove points in the set P of        “pathological” nodes based on user input.    -   4) While P is not empty:        -   a. For each node (p) of P, apply a “smoothness” constraint,            e.g., defined by an equation of (p) specifying that, in a            neighborhood of (p), the geological-time function t(x,y,z)            does not have a maximum or minimum;        -   b. Generate a subsequent geological-time function t(x,y,z)            at the nodes of a given mesh M₁ by approximating the            geological-time function t(x,y,z) with the smoothness            constraint and other constraints generated by other sources            of information. For example, the DSI method or any other            equivalent approximation method able to take into account            smoothness constraints may be used.        -   c. Re-compute the set P of “pathological” nodes of the mesh            where there is a local maximum or minimum of t(x,y,z). If P            is not empty, repeat step 4.    -   5) Stop

In step (3a) above, a “smoothness” constraint may be used, defined, forexample, by one or more of the following (other definitions may be usedin other embodiments):

-   -   Smoothness constraint #1: In the neighborhood of (p), the        gradient of t(x,y,z) is constrained to be as constant as        possible;    -   Smoothness constraint #2: In the neighborhood of (p), the        function t(x,y,z) is constrained to vary linearly, as much as        possible;    -   Smoothness constraint #3: In the neighborhood of (p), the        function t(x,y,z) is constrained to be as close as possible to a        harmonic function.

Removing pathological nodes (e.g., bubbles) may distort the resultinggeological-time function t(x,y,z) and, consequently, the time functionmay be so smooth that it fits the associated geological dataincorrectly. In such cases, the resulting geological-time functiont(x,y,z) may be set to be a first approximation t₀(x,y,z) of a correctedor refined geological-time function t(x,y,z) that better fits the data.It may be appreciated that by removing bubbles, the gradient of thefirst approximation geological-time function t₀(x,y,z) typically nevervanishes and its gradient G₀(x,y,z) may be approximately orientedorthogonally to the reference horizons fit by a correctedgeological-time function t(x,y,z). As a result, the angle between thegradient G(x,y,z) of the corrected geological-time function t(x,y,z) andthe gradient G₀(x,y,z) of the first approximation geological-timefunction t₀(x,y,z) may be small, for example, less than a predeterminedvalue. Based on this relationship, the following post-processing stepsmay be applied to refine a geological-time function t(x,y,z) that doesnot fit the data correctly (other operations or series of operations maybe used, and the exact set of steps shown below may be varied):

-   -   1. Set an initial geological-time function t(x,y,z) obtained        after removing bubbles to be equal to a first approximation        t₀(x,y,z) of the geological-time function;        a. t ₀(x,y,z)=t(x,y,z) for all (x,y,z)    -   2. At any location in the studied geological domain:        -   a. compute the gradient G₀(x,y,z) of t₀(x,y,z);        -   b. apply the following inequality constraint:            G(x,y,z)·G ₀(x,y,z)>∥G ₀(x,y,z)∥²·cos(A) for all (x,y,z),            where the symbol “·” represents the dot product operation            between the gradient G(x,y,z) of a corrected geological-time            function t(x,y,z) and the gradient G₀(x,y,z) of the first            approximation geological-time function t₀(x,y,z).

In the equation of step (2b) above, A may be a predetermined maximumangle between the gradients G(x,y,z) and G₀(x,y,z). For example, angle Amay be between [−60,−60] degrees.

The equation of step (2b) above may be linear relatively to thecorrected geological-time function t(x,y,z), and accordingly, may beapplied using a DSI inequality constraint. In one embodiment, thefollowing post-processing steps may be applied, if needed, after theapplication of the bubble removal steps:

-   -   (I) Set the first approximation t₀(x,y,z) to be equal to the        resulting geological-time function t(x,y,z) obtained after        bubble removal steps are applied;    -   (II) To compute the corrected geological-time function t(x,y,z),        add the inequality constraint of step (2b) above to the process        described in the section entitled “Unified Modeling of a        Geological-Time Function Integrating Multiple Sources of        Information.”        Controlling the Quality of the Geological-Time Function

Once the geological-time function, t(x,y,z), is defined, the accuracy ofthe function may be measured and, if it is insufficient, thegeological-time function, t(x,y,z) may be re-calculated and refined. Inone embodiment, input data used for deriving the geological-timefunction, t(x,y,z), may be primarily composed of control-points(x_(i),y_(i),z_(i)) at which the geological-time function is defined.

To control the quality of the geological-time function, t(x,y,z), of amodel defined on a mesh M₁, the following process may be used (otheroperations or series of operations may be used, and the exact set ofsteps shown below may be varied).

-   -   1) For each control-point (x_(i),y_(i),z_(i)):        -   a. Retrieve the coordinate location (x_(i),y_(i),z_(i)) of            the control-point and the corresponding value (t_(i)) of the            geological-time function.        -   b. Using values of t(x,y,z) previously computed at nodes of            the mesh M₁, locally approximate the value            t(x_(i),y_(i),z_(i)) of the geological-time at the            control-point (x_(i),y_(i),z_(i)).        -   c. Compute G_(i), which is the module of the gradient of            t(x,y,z) at location (x_(i),y_(i),z_(i)).        -   d. Compute the error E_(i)=(t(x_(i),y_(i),z_(i))−t_(i))        -   e. Compute the absolute value of E_(i)/G_(i) and set it            equal to D_(i) considered as an estimate of the distance            between the control-point and the horizon H(t_(i)) on which            it should be located    -   2) Choose a 2D or 3D graphical user interface on a display        device (e.g., display 180 of FIG. 10) and a color map, e.g.,        defining a one-to-one correspondence between a color C_(i) and a        distance D_(i)    -   3) Display each control-point (x_(i),y_(i),z_(i)) on the        graphical display device with the colored symbol C_(i) of the        color map corresponding to the distance D_(i). For example, a        control point may be represented by a small cube or a small        sphere.    -   4) Stop

Embodiments of the invention may include determining the accuracy of ageological-time function, t(x,y,z), of a model. An accuracy test isexecuted for the geological-time function, t(x,y,z), at each of aplurality of points, e.g., control points (x_(i),y_(i),z_(i)).

In one embodiment of the invention, the geological-time function isknown at nodes of mesh M₁. The known values of the geological-timefunction at the nodes (x,y,z) are approximated to determine thecorresponding values of the geological-time function at control points(x_(i),y_(i),z_(i)). The difference between the value t, of thegeological-time function and the value approximated based on neighboringnodes is determined. If the difference is greater than a predeterminedthreshold then the geological-time function may be inaccurate. Eachpoint at which this difference is greater than a predetermined thresholdmay be re-calculated by a processor repeating the steps above.Alternatively or additionally, if the difference between thegeological-time function and the approximated corresponding values, forall points or a set of points in the model, is on average or in totalgreater than a predetermined threshold, the geological-time function ofthe entire model may be re-calculated.

Reconciling Seismic and Well Data

Typically, a set of control-points sampled on reference horizons may beautomatically split into distinct subsets, for example, as follows:

-   -   a subset, control-points derived from seismic cross sections or        seismic cubes, (CPS), corresponding to points extracted on        reference horizons observed on seismic cross sections or seismic        cubes; and    -   a subset, control-points observed on well paths, (CPW)        corresponding to well markers at the intersection of well paths        with reference horizons.

These set of control-points, sampled from the same geological horizons,but with respect to different sources of information, are typicallydistinct and have complementary properties:

-   -   The seismic data corresponding to the set CPS is typically dense        but the vertical positioning of reference horizons is often        unknown because of geological uncertainties.    -   The well data corresponding to the set CPW typically provides        precise positioning of the horizons but the set is very sparse        in the studied domain.

As a consequence, there is generally a discrepancy between thepositioning of the reference horizons when control-points from the CPSand CPW sets are used. Embodiments of the invention propose thefollowing steps to reconcile this discrepancy.

In one embodiment of the invention, seismic data corresponding to theset CPS may be used to determine the shape of the reference horizons,while the well data corresponding to the set CPW may be used todetermine the position of the reference horizons. Accordingly, a processmay proceed as follows (other operations or series of operations may beused, and the exact set of steps shown below may be varied).

-   -   1) The geological domain of interest may be covered by a        corresponding mesh M₁. In a preferred embodiment, the cells of        M₁ may be tetrahedral cells (e.g., as shown in FIG. 6) or        hexahedral cells, although other types of cells or polyhedral        cells may be used. The mesh M₁ may be generated, as is known in        the art.    -   2) The geological-time function t(x,y,z) may be fully defined by        its values at the vertices of mesh M₁. The geological-time        function may be generated at an arbitrary point (x,y,z) by        locally approximating known values of the geological-time        function sampled at the nodes of M₁ neighboring the point        (x,y,z).    -   3) A first approximation t_(s)(x,y,z) of the geological-time        function may be generated using the seismic data set CPS only        and discarding the well data set CPW.    -   4) “Control-points” constraints may be applied to specify that,        for each sampling point (x,y,z) in the set CPW and located on a        reference horizon H(t_(i)), t(x,y,z)=t_(i) where t_(i) is the        geological-time of H(t_(i)).    -   5) Choose a dense set of points CG in the studied domain. These        points may correspond to points where the final solution has the        same shape (but not necessarily the same position) as the        solution provided by the points of the set CPS. For example but        not limited to, one can choose the centers of the cells of M₁.    -   6) For each point (x_(g),y_(g),z_(g)) of CG:        -   a. Compute the gradient Gs(x_(g),y_(g),z_(g)) of            t_(s)(x,y,z) at location (x_(g),y_(g),z_(g)). The direction            of the gradient Gs(x_(g),y_(g),z_(g)) is assumed to            characterize the shape of the horizon going through            (x_(g),y_(g),z_(g)).        -   b. apply a constraint specifying that the gradient            G(x_(g),y_(g),z_(g)) of t(x,y,z) at location            (x_(g),y_(g),z_(g)) may be, approximately, parallel to the            gradient Gs(x_(g),y_(g),z_(g)).    -   7) Constraints may be used which specify that, over the studied        domain, the geological-time function t(x,y,z) be as smooth as        possible.    -   8) For each fault F in which the type of fault, e.g., a normal        fault or a reverse fault, is known:        -   a. sample the fault F with a set of collocated points            n_(i)=n(x_(i),y_(i),z_(i)) on the negative side of the fault            F and p_(i)=p(x_(i),y_(i),z_(i)) on the positive side of the            fault F;        -   b. for each pair of sampling points (n_(i),p_(i)), according            to the type of fault of fault F, apply the appropriate one            of the following inequality constraints:            -   If fault F is a normal fault, then add the constraint                t(p_(i))>t(n_(i))            -   If fault F is a reverse fault, then add the constraint                t(p_(i))<t(n_(i))            -   The inequality constraints may be, for example, “Delta”                constraints in the DSI method or any other equivalent                interpolation or approximation method able to take into                account linear equality and inequality constraints.    -   9) The DSI method or any other equivalent approximation method        may be applied to compute the values of the geological-time        function t(x,y,z) at the vertices of mesh M₁ so that the        constraints defined above are honored. The multiple constraints        may be combined and simultaneously applied to calculate the        geological-time function, as described above.    -   10) Stop

In step (7) above, a “smoothness constraint” may be used, defined, forexample, by one or more of the following (other constraints may beused):

-   -   Smoothness constraint #1: In the neighborhood of each node (n)        of M₁, the gradient of t(x,y,z) is constrained to be as constant        as possible;    -   Smoothness constraint #2: In the neighborhood of each node (n)        of M₁, the function t(x,y,z) is constrained to vary linearly, as        much as possible;    -   Smoothness constraint #3: In the neighborhood of each node (n)        of M₁, the function t(x,y,z) is constrained to be as close as        possible to a harmonic function.

Each of these smoothness constraints is preferably linear so that theconstraint may be easily implemented in the DSI method or any otherequivalent approximation method to further constrain the values of thegeological-time function t(x,y,z) at each collocated point n_(i) of meshM₁. Alternatively, the constraints may be non-linear and may beapproximated as a linear function or used with an algorithm enablingnon-linear constraints.

According to an embodiment of the invention, control-points from a setof seismic data, CPS, and a set of well data, CPW, are used by a processto generate different reference horizons. In one embodiment of theinvention, control-points from a set of seismic data CPS may be used todetermine the shape of the reference horizons and control-points from aset of well data CPW may be used to determine the position of thereference horizons with similar shape (but different positioning) asthose generated from the set CPS. Alternatively, other or overlappingsets of CPS and CPW may be used to generate different reference horizonsand to determine the shape of the reference horizons.

Modeling Uncertainties on Horizons

In sedimentary geology, a horizon may be considered to be a surfaceincluding a set of particles of sediment that were uniformly depositedat the same geological-time. As with other subsurface features, ahorizon may be represented in a computer system as data.

-   -   Each horizon should be a level surface onto which the        geological-time function is constant.    -   If the geological-time function has a local minimum or maximum,        then there exists mandatorily a closed level surface around that        local minimum or maximum. From a geological perspective, closed        surfaces cannot represent valid horizons: therefore, the        geological time function should not have any local maximum or        minimum.

Note that a geological-time function does not need to represent theexact geological-time: it simply needs to be an increasing function ofthe real (generally unknown) geological-time

Conventional mechanisms for modeling uncertainties on horizons such asthe “GeoChron” modeling mechanism described in the background of thisinvention, or any other equivalent mechanism, may yield “bubbles” orgeological-time functions t(x,y,z) with local minima and maxima for agiven stochastic or uncertain occurrence (m). From a geologicalperspective, these minima and maxima are undesirable and may be removed.To remove these minima and maxima while minimally modifying the geometryof the horizons corresponding to level surfaces of t(x,y,z), thefollowing operations are proposed (other operations or series ofoperations may be used, and the exact set of steps shown below may bevaried):

-   -   1. Assume that a stochastic occurrence (m) of a geometrical        model of a horizon is represented by a given geological-time        function t(x,y,z|m) defined by its values at the nodes of a        given 3D mesh M₁. The geological-time function t(x,y,z|m) may be        built using a conventional algorithm.    -   2. At the nodes of mesh M₁, define a new function t₀(x,y,z).    -   3. Compute or find a set P of “pathological” nodes of the mesh        M₁, e.g., where t(x,y,z|m) is locally minimum or maximum.    -   4. Let Q be an empty set of nodes. For each node (p) of P, add        to Q the nodes of M₁ within a given distance or neighborhood of        (p).    -   5. For each node (n_(i)) of mesh M₁ not in Q (e.g., represented        by coordinates (x_(i),y_(i),z_(i))), a constraint may be used        specifying that the value of t₀(x_(i),y_(i),z_(i)) may be        approximately equal to t(x_(i),y_(i),z_(i)), for example in a        least squares sense. That is, a new value of the pathological        node or region may be estimated from the value of other nodes in        the mesh. For example, a “soft” (or “fuzzy”) control node/point        may be used.    -   6. While P is not empty, repeat the following steps (a) to (c):        -   a. For each pathological node (p) of P, model t₀(x,y,z) by            adding a local “smoothness” constraint at (p), for example,            specifying that, in a neighborhood of (p), the function            t₀(x,y,z) may not have any maximum or minimum value.        -   b. Run an approximation mechanism (e.g., using a known            approximation method able to take into account a local            smoothness constraint) to generate the function t₀(x,y,z).        -   c. Compute an updated set P of “pathological” nodes of the            mesh M₁ where t₀(x,y,z) is locally minimum or maximum.    -   7. For each node (n_(i)) (with coordinates (x_(i),y_(i),z_(i)))        of the mesh M₁, replace value t(x_(i),y_(i),z_(i)|m) by value        t₀(x_(i),y_(i),z_(i)).    -   8. Stop

A “smoothness” constraint may include any requirement used to preventt₀(x,y,z) from having a local maximum or minimum value at a pathologicalnode p. For example, smoothness” constraint may include:

-   -   Smoothness constraint #1: In the neighborhood of (p), the        gradient of t₀(x,y,z) is constrained to be as constant as        possible;    -   Smoothness constraint #2: In the neighborhood of (p), the        function t₀(x,y,z) is constrained to vary linearly, as much as        possible;    -   Smoothness constraint #3: In the neighborhood of (p), the        function t₀(x,y,z) is constrained to be as close as possible to        a harmonic function.

As with other processes described herein, it is contemplated thatmodifications and combinations of this algorithm will readily occur tothose skilled in the art, which modifications and combinations will bewithin the spirit of the invention.

Other constraints may be used.

A geological-time function represents the time when particles ofsediment were originally deposited. A geological-time function need notrepresent the exact geological-time but may be an increasing function ofthe real (generally unknown) geological-time. Since each horizonrepresents a set of particles that were uniformly deposited in time (atthe geological-time of deposition), each horizon should be a levelsurface on which the geological-time function is constant. Conventionalmechanisms are known to generate geological-time functions having localmaximum or minimum values, thereby violating the requirement that thehorizons correspond to level surfaces of the geological-time function.

In one embodiment of the invention, a geological-time function,t(x,y,z), may be generated without (or with a substantially reducednumber of) local maximum or minimum values in the 3D studied domain.That is, the geological-time function, t(x,y,z), may be substantiallyconstant on a horizon, and substantially monotonic across differenthorizons.

In one embodiment of the invention, a computing module may be executed(for example, by a processor as described herein) to identify a firstset of nodes having a local maximum or minimum value for thegeological-time function, t(x,y,z), in a first geological model. Thecomputing module may be executed to generate a second geological modelhaving a second set of nodes corresponding to the first set of nodes.The second set of nodes may duplicate the first set of nodes oralternatively, may be rearranged. The modeling module may be executed tocompute values for the geological-time function, t(x,y,z), of the secondset of nodes that are different than the corresponding values for thefirst set of nodes. The new values for the geological-time function,t(x,y,z), are preferably not local maxima or minima values in the 3Dmodeled domain. For example, in order to generate a level horizon with asubstantially constant geological-time function, the new values maysubstantially match the values of neighboring nodes in the horizon. Inone example, the modeling module may be executed to compute the valuefor the geological-time function, t(x,y,z), of the second set of nodesby duplicating, approximating, or otherwise using the values of nodes inthe first model that neighbor the nodes having a local maximum orminimum value. Once a new set of values for the geological-timefunction, t(x,y,z), are defined which do not have local maxima or minimavalues in the 3D domain, the new set of values may replace the originalset of values in the first model for corresponding nodes. To generate asubstantially continuous new geological-time function, t(x,y,z), valuesof this function at neighboring nodes may also be replaced by themodeling module to smooth differences in the geological-time function,t(x,y,z), between the initial and new values. The resultinggeological-time function, t(x,y,z), with new substantially continuousvalues replaces any problematic local maxima and minima values in theinitial model, thereby improving the accuracy of the model.

Other data or combinations of data and mechanisms to manipulate the datamay be used by the modeling module to execute embodiments of theinvention.

Modeling Uncertainty Related to the Positioning of Particles of Sediment

Seismic data may have multiple sources of uncertainty. For example, asdiscussed in the section entitled “Geometrical Uncertainties andPreferred-Model”, seismic data includes inherent geologicaluncertainties such as the location of sediment particles and faults.Furthermore, conventional processing steps often rely on approximatecalculations to process the seismic data adding computing uncertaintiesto the already geologically uncertain data. These computinguncertainties include, for example, errors induced by seismic processingand errors on variations of the seismic velocity field used in the timeto depth conversion in seismic processing algorithms. Not all sucherrors may be present in models processed by embodiments of theinvention, and other errors may be present.

Embodiments of the invention describe a system and method for perturbingan initial 3D geological model using a 3D vector field. A processor(e.g., processor 140 of FIG. 10) may generate a 3D vector fieldincluding 3D vectors. Each 3D vector of the 3D vector field may beassociated with a node of the initial 3D geological model and has amagnitude within a range of uncertainty of the node of the initial 3Dgeological model associated therewith. The processor may apply the 3Dvector field to the initial 3D geological model associated therewith togenerate an alternative 3D model. The alternative 3D model may differfrom the initial 3D geological model by a 3D vector at the nodes havinguncertain values.

For any equiprobable stochastic occurrence (m) of the geometric model,the uncertainties of the seismic data may be modeled by a vector fieldv(x,y,z|m). The vector field v(x,y,z|m) for each instance of anequiprobable occurrence, (m), may be applied to the initial model G_(M)(e.g., a “preferred model”) to cause each node (e.g., modeling thelocation of a particle of sediment or other feature) located at point(x,y,z) in the initial model G_(M) to move to a new (e.g., stochastic)location (x,y,z|m) in the perturbed model GM(m), for example, asfollows:(x,y,z|m)=(x,y,z)+v(x,y,z|m)  [7]

To generate a coherent perturbation of the initial model, the vectorfield v(x,y,z|m) is preferably substantially continuous thereby avoidinggenerating voids or collisions between points of the model. Tounderstand the coherency problem which may arise, replace equation [7]by equation [7a], for example, as follows:(x,y,z|m)=(x,y,z)+s·v(x,y,z|m)  [7a]

where “(s·v)” represents the product of a scalar (s) by a vector (v) andwhere (s) is a scaling parameter in range [0,1]. When the vector fieldsare defined at a finite number of nodes, the vector fields being onlyevaluated at each incremental node, are typically not themselvescontinuous, but may approximate or coincide with a continuous vectorfield.

Consider an infinitely small element of volume ΔVol(x,y,z) initiallycentered on location (x,y,z) of the initial model, the internal pointsof which are perturbed according to equation [7a]. When (s) variescontinuously from 0 to 1, the volume ΔVol(x,y,z) may also varycontinuously from an initial geometry in the initial model to aperturbed geometry in the perturbed model (e.g., the same results may beobtained by applying the perturbation v(x,y,z|m) in equation [7]directly). However, if, during such a continuous transformation, thevolume ΔVol(x,y,z) vanishes for some value s=s₀ in [0,1], then thegeologic material contained initially within ΔVol(x,y,z) vanishes forthat particular value s₀ of s. This is a physical impossibility.Therefore, coherent perturbations may preferably be defined such that,regardless of shape and dimension, an initial volume ΔVol(x,y,z) doesnot vanish when the volume is perturbed. Similarly, a perturbation by a3D vector field may be coherent if and only if any infinitely smallpolyhedron whose vertices are translated according to this vector fieldis transformed in such a way that each of its vertices never crosses anyof the polyhedron faces. For example, if deformations induced by vectorfield v(x,y,z|m) are small, it has been proven by the first inventor ofthis patent application that to ensure the coherency of the perturbationinduced by vector field v(x,y,z|m), it suffices that the divergence ofthe vector field v(x,y,z|m) (e.g., div {v(x,y,z|m)=dvx/dx+dvy/dy+dvz/dz)is bounded to be greater than (−1) and less than (3/2) in the studieddomain:

$\begin{matrix}{{- 1} < {{div}\left\{ {v\left( {x,y,\left. z \middle| m \right.} \right)} \right\}} < \frac{3}{2}} & \lbrack 8\rbrack\end{matrix}$

If the vector field v(x,y,z|m) is sampled at the nodes of a meshcovering the 3D space, v(x,y,z|m) may be approximated by a linearcombination of the values of v(x_(i),y_(i),z_(i)|m) corresponding to thevalues of v(x,y,z|m) at the nodes of the mesh in the neighborhood of(x,y,z). The divergence is a linear operator. To generate a “coherency”constraint specifying that v(x,y,z|m) be defined by equation [7], forexample, a finite difference linear approximation of the derivativesoccurring in the definition of the divergence operator may be used.

For initial models having an uncertain property (e.g., the location ofwell markers or faults), embodiments of the invention include perturbingthe initial model to generate and display a plurality of perturbedmodels, each perturbed model having a different arrangement of nodes,each arrangement having substantially the same probability ofoccurrence. Each perturbed model may be generated by applying acorresponding vector field. The applied vector fields typically causethe nodes of the initial model to move to corresponding new locations inthe perturbed models. The vector fields are preferably substantiallycontinuous (e.g., having few or no points of discontinuity). When thevector fields are defined at a finite number of nodes, the vector fieldsbeing only evaluated at each incremental node, are typically notthemselves continuous, but may approximate or coincide with a continuousvector field. Discontinuous vector fields (e.g., coinciding with vectorfields at points at which the vector fields are discontinuous) maythemselves induce addition errors. In one example, when using adiscontinuous vector field, initially neighboring nodes may be moved tonon-neighboring locations, generating erroneous gaps in the perturbedmodel. In another example, discontinuous vector fields may move thenodes across faults or to collide with each other. By usingsubstantially continuous vector fields to move nodes of the initialmodel, the perturbed model may avoid such errors induced by the vectorfields themselves.

However, even when substantially continuous vector fields are applied tothe initial model, uncertainties (e.g., errors) in the data of theinitial model may carry over to the perturbed model. For example, avolume or neighborhood of the initial model may shrink, vanish to apoint or reverse signs, when the substantially continuous vector fieldsare applied. The vector field is constrained so that, when applied topoints of an initial positive volume, the volume never vanishes. In oneembodiment of the invention, vector fields may be constrained by afunction of the divergence of the vector fields. As shown in equation[8], the divergence of each vector field is bounded within a range of

$\left( {{- 1},\frac{3}{2}} \right).$Vector fields may be constrained by a function of the divergence of thevector fields, for example, by a computing module executing stepsaccording to the description in the section entitled “Testing andCorrecting Perturbations Vector Fields”. Other constraints may be usedto bound the vector fields to generate accurate perturbations of theinitial model.Testing and Correcting Perturbations Vector Fields

The plurality of vector fields {v(x,y,z|m₁), v(x,y,z|m₂, . . . ,v(x,y,z|m_(n)} may be tested and corrected to ensure the aforementionedcoherence constraints are satisfied, for example, by a computing moduleexecuting the following steps (other operations or series of operationsmay be used, and the exact set of steps shown below may be varied):

Testing Phase

1. Cover a 3D geological domain of interest by a mesh M₂. Mesh M₂ ispreferably continuous across faults. M₂ may be a regular mesh havingsubstantially identically shaped (e.g., hexahedron or tetrahedron)cells.

2. Select a random number (m) to be used as a “seed” by a randomfunction generator.

3. Use a stochastic simulator to generate an equiprobable occurrencev(x,y,z|m) of a perturbation vector field by any known method:

4. Compute the divergence of v(x,y,z|m) at each node or on each cell ofthe mesh M₂.

5. Determine:

-   -   a. If the divergence is greater than (−1) and lower than (3/2)        at each node or on each cell of the mesh M₂, proceed to step (6)    -   b. Otherwise select one of the following:        -   i. discard the occurrence v(x,y,z|m), or        -   ii. proceed to step (1) of the “Correcting Phase” below to            correct v(x,y,z|m).

6. If a subsequent equiprobable occurrence v(x,y,z|m) exists, return tostep (2) using this occurrence.

7. Stop

When the vector field v(x,y,z|m) is determined to violate the coherenceconstraint (e.g., in step 5b), the vector field may be discarded orcorrected, e.g., according to the mode of operation of a computingmodule executing the aforementioned steps to either discard or correctthe erroneous vector field. For example, a programmer or user may set amode of operation of the computing module to establish how to resolvesuch an erroneous vector field v(x,y,z|m) by either discarding orcorrecting the vector field.

Alternatively, the mode of operation to either discard or correct theerroneous vector field may be automatically selected in the computingmodule. In one embodiment, the selected mode of operation may depend onthe extent to which the vector field v(x,y,z|m) violates the coherenceconstraint. For example, if a vector field violates the coherenceconstraint at a single node or for a sufficiently small number of nodesand/or is within a predetermined acceptable range of error, the vectorfield may be weighted, ranked, or demoted accordingly. In oneembodiment, the plurality of perturbed models may be displayed in anorder corresponding to the rank of the vector field with which they weregenerated. In such embodiments, the computing module need not executeadditional operations to correct each error of a vector field, but may,e.g., indicate by a marking of rank or the order with which the model isdisplayed, which models are correct, incorrect, and partially correct.For example, a first model displayed to a user may be generated by avector field in which every node of the vector field satisfies thecoherence constraint. The last model displayed to a user may begenerated by a vector field in which few nodes satisfy the coherenceconstraint. There may for example be a maximum acceptable threshold ofdeviation beyond which vector fields are discarded.

In one embodiment of the invention, each vector field, which is appliedto the initial model to generate a corresponding perturbed model mayitself be displayed (e.g., on display 180 of FIG. 10), e.g., separatelyfrom or adjacent to the corresponding perturbed model.

Correcting Phase

When a vector field v(x,y,z|m) is determined to violate a coherenceconstraint, as in step 5b(ii) above, the vector field v(x,y,z|m) may becorrected to satisfy the constraint, for example, by the following stepsexecuted by a computing module (other operations or series of operationsmay be used, and the exact set of steps shown below may be varied):

-   -   1. Accept a vector field v(x,y,z|m) determined to violate a        coherence constraint.    -   2. Create a copy v₀(x,y,z|m) of the vector field v(x,y,z|m)    -   3. For each node (n) of mesh M₂, determine the coordinate        location (x_(n),y_(n),z_(n)) and apply a constraint specifying        that a new value of v(x_(n),y_(n),z_(n)|m) interpolated at this        location may be approximately equal to v₀(x_(n),y_(n),z_(n)|m),        for example, in a least squares sense. For example, a “soft” (or        “fuzzy”) control node may be used.    -   4. At each node of M₂ or on each cell of M₂, apply a constraint        specifying that new values of v(x,y,z|m) satisfy the inequality        [8]. For example, the DSI method (e.g., as described in [MAL02]        and [MAL08]) or any other equivalent approximation method may be        used to take these inequalities into account as two inequality        constraints.    -   5. A smoothness constraint may be applied specifying that the        new values of v(x,y,z|m) be as smooth as possible in the modeled        domain.    -   6. For example, the DSI method or any other equivalent        approximation method may be used to compute the new values of        v(x,y,z|m) at the nodes of mesh M₂ while satisfying the        constraints above.    -   7. Stop

In step (5) above, a “smoothness constraint” may be used, defined in theneighborhood of each node (n) of mesh M₂, for example, by one or more ofthe following:

-   -   Smoothness constraint #1: In the neighborhood of (n), the        gradient of the x, y and z components of v(x,y,z|m) are        constrained to be as constant as possible;    -   Smoothness constraint #2: In the neighborhood of (n), the x, y        and z components of v(x,y,z|m) are constrained to vary linearly,        as much as possible;    -   Smoothness constraint #3: In the neighborhood of (n), the x, y        and z components of v(x,y,z|m) are constrained to be as close as        possible to a harmonic function.        Generating a Vector Field of Coherent Perturbations Associated        with a Fault

Faults or unconformities are generally observed and sampled on seismiccross sections or automatically extracted from seismic cubes. Sincefaults often appear “fuzzy” on seismic cross sections, the exactlocation of faults may be uncertain and thus, it may be difficult toposition faults at an exact location on a 3D mesh M₁ of a geologicalmodel.

According to embodiments of the invention, when the position of a faultis uncertain, a plurality of vector fields may be generated, each ofwhich may be applied to an initial model of the fault, which in turngenerates a plurality of perturbed fault models. The applied vectorfields may be generated for example, by the following steps executed bya computing module (other operations or series of operations may beused, and the exact set of steps shown below may be varied).

-   -   1. Cover a 3D geological domain of interest by a mesh M₂. Mesh        M₂ is preferably continuous across faults. M₂ may be a regular        mesh having substantially identically shaped (e.g., hexahedron        or tetrahedron) cells.    -   2. Select a random number (m) to be used as a “seed” by a random        function generator.    -   3. Generate a vector field v(x,y,z|m) to be built in the 3D        studied domain for each equiprobable stochastic occurrence (m)        such that:        -   a. v(x,y,z|m) is defined by sampling values at the nodes of            mesh M₂,        -   b. at any location (x,y,z) in the 3D studied domain, the            value of v(x,y,z|m) may be locally approximated from the            values sampled at the nodes of mesh M₂ in the neighborhood            of (x,y,z).    -   4. Using a stochastic generator known in the art, generate an        equiprobable occurrence v_(F)(x,y,z|m) of a vector field        modeling the equiprobable stochastic occurrence (m) of the        uncertainty at point (x,y,z) defined on the surface of fault F.        The vector field v_(F)(x,y,z|m) may be generated at each of a        set of given sampling points S_(F)={(x₁,y₁,z₁), . . . ,        (x_(q),y_(q),z_(q))} located on fault F. Any stochastic        simulator or processing module may be used to generate the        equiprobable occurrence v_(F)(x,y,z|m).    -   5. For each sampling location (x_(f),y_(f),z_(f)) in set S_(F),        apply a constraint specifying that the value of        v(x_(f),y_(f),z_(f)|m) approximated at location        (x_(f),y_(f),z_(f)) may be approximately equal to        v_(F)(x_(f),y_(f),z_(f)|m), for example in a least squares sense        but not limited thereto. For example, a “soft” (or “fuzzy”)        control node/point may be used.    -   6. For each node (n) of M₂ having a coordinate location        (x_(n),y_(n),z_(n)) greater than a predetermined distance from        fault F, apply a constraint specifying that the value of        v(x_(n),y_(n),z_(n)|m) approximated at location        (x_(n),y_(n),z_(n)) may be approximately equal to a null vector,        for example in a least squares sense but not limited thereto.        For example, a “soft” (or “fuzzy”) control node/point may be        used.    -   7. If F is a secondary fault which branches from a main fault        F_(main) along a common border B, then:        -   a. Build a set S_(B) of sampling points close to B; and        -   b. For each point b of S_(B) whose coordinates are            (x_(b),y_(b),z_(b)), apply a constraint specifying that the            dot product of v(x_(b),y_(b),z_(b)|m) with the normal vector            of F_(main) at location (x_(b),y_(b),z_(b)) vanishes (i.e.,            approximates zero (0)). Accordingly, the secondary faults            are perturbed in a direction of the main fault F_(main) and            move along the main fault F_(main) without altering the            general fault shape. Such a constraint is linear and may be            taken into account by the DSI method or another            approximation method.    -   8. At each node of M₂ or on each cell of M₂, apply a constraint        on v(x,y,z|m) specifying that inequality [8] is satisfied. For        example, if the DSI method is used to approximate v(x,y,z|m),        then the constraint [8] may be expressed as two inequality        constraints.    -   9. Smoothness constraints may be applied to specify that        v(x,y,z|m) should be as smooth as possible.    -   10. For example, the DSI method or any other equivalent        approximation method may be used to compute the new values of        v(x,y,z|m) at the nodes of mesh M₂ while satisfying the        constraints above.    -   11. If a subsequent equiprobable occurrence (m*) exists, return        to step (2) to generate a subsequent v(x,y,z|m*) with the        occurrence (m*).    -   12. Stop

In step (9) above, a “smoothness constraint” may be used, defined in theneighborhood of each node (n) of mesh M₂, for example, by one or more ofthe following (other constraints may be used):

-   -   Smoothness constraint #1: In the neighborhood of (n), the        gradient of the x, y and z components of v(x,y,z|m) are        constrained to be as constant as possible;    -   Smoothness constraint #2: In the neighborhood of (n), the x, y        and z components of v(x,y,z|m) are constrained to vary linearly,        as much as possible;    -   Smoothness constraint #3: In the neighborhood of (n), the x, y        and z components of v(x,y,z|m) are constrained to be as close as        possible to a harmonic function.

To alter the initial model to generate each of a plurality of perturbedmodels, vectors of the vector field v(x,y,z|m) may be applied tocorresponding nodes of the initial model. According to embodiments ofthe invention, the vector field v(x,y,z|m) is three-dimensional. Thatis, the vectors of the vector field v(x,y,z|m) may have a differentdirection in a 3D space.

Accordingly, when the three-dimensional vector field v(x,y,z|m) isapplied to the initial model, nodes of the initial model may be moved indifferent directions in the 3D space. In conventional methods, nodes ofthe initial model were moved in the same directions (i.e., restricted to1 dimension) of the 3D space. As an unfortunate consequence, if a firstfault F is moved, which initially branches on a second fault, theperturbed faults are likely to have a geologically unacceptable shapeclose to the branching. According to the process described, since eachnode may be moved in a different direction, the first fault may be movedin continuous parts. For example, nodes close to or within a smallpredetermined neighborhood of the branching of the first fault and thesecond fault may be moved in the direction of the second fault, slidingalong the length of the second fault, so that the first and secondfaults remain smooth. The direction of displacement for the remainingportion of the first fault varies continuously from the direction of thesecond fault to the chosen displacement direction for the first faultfurther away from the branching. Using a smoothly varying vector fieldv(x,y,z|m) to move individual nodes of an initial model in differentdirections may preserve the relative structures of the initial model,e.g., such as networks of branching faults, while executing randomperturbations of the remainder of the model.

Applying a vector field v(x,y,z|m) to corresponding nodes of the initialmodel may cause corresponding nodes in the perturbed model to “move”,“slide”, or “rotate” relative thereto, thereby transforming modeledgeological features represented by nodes and cells of the initial modelto a new location in the perturbed model.

Generating a Vector Field of Coherent Perturbation Associated withSeismic Velocity

The location (x,y,z) of a particle of sediment in a geological model(e.g., the node, cell, vertex, or coordinate location) may be determinedusing an approximation of seismic velocity S(x,y,z). Seismic velocityS(x,y,z) typically has a significant amount of uncertainty, which inturn causes the location (x,y,z) of the particle of sediment to bemodeled with uncertainty.

An error function, r(x,y,z|m), may be defined. The error function,r(x,y,z|m), may represent relative error of the seismic velocityS(x,y,z) at location (x,y,z) for an occurrence (m) of the geometricmodel. The error function, r(x,y,z|m), may represent the relative errorof the seismic velocity, for an occurrence (m), for example, as follows:r(x,y,z|m)=ΔS(x,y,z|m)/S(x,y,z)  [9]

where S(x,y,z) is a measure of seismic velocity and ΔS(x,y,z|m) is ameasure of the error associated with the seismic velocity S(x,y,z).Typical values for the magnitude of the relative error of the seismicvelocity are, for example, in a range of from approximately ±3% toapproximately ±10%. The uncertainty of seismic velocity may, forexample, depend on the depth (z) associated with the location (x,y,z).

u(x,y,z) may represent a unit vector substantially tangential to aseismic ray passing through (x,y,z); h(x,y,z) may represent a thicknessin direction u(x,y,z) of a thin layer passing through (x,y,z); andΔh(x,y,z|m) may represent an error associated with the thicknessh(x,y,z). Accordingly, the error function, r(x,y,z|m), may alsorepresent the relative error of the thickness, which may for example berepresented as follows:Δh(x,y,z|m)/h(x,y,z)=r(x,y,z|m)  [10]

Combining equation [9] and equation [10], an equality is establishedbetween the relative error of the seismic velocity and the relativeerror of the thickness of a layer. Accordingly, adding a (positive ornegative) random error ΔS(x,y,z|m) to the seismic velocity S(x,y,z) isequivalent to moving the particle of sediment (x,y,z) in directionu(x,y,z) to a new location (x,y,z|m). The particle of sediment (x,y,z)may be moved, for example, according to the following equation:(x,y,z|m)=(x,y,z)+v(x,y,z|m)  [11a]where v(x,y,z|m)=e(x,y,z|m)·u(x,y,z)  [11b]

where the symbol “(e·u)” represents the product of a scalar (e) by avector (u) and where e(x,y,z|m) is a scalar function of random error at(x,y,z) for an occurrence (m). The first inventor of this patentapplication proved that:

-   -   the error e(x,y,z|m) may be determined for example, according to        equation [12] as follows.        grad e(x,y,z|m)·u(x,y,z)=r(x,y,z|m)  [12]    -   where the symbol “·” represents the dot product operation        between two vectors, grad(e) and u, and where the dot product of        the gradient {grad e(x,y,z|m)} of the error e(x,y,z|m) and the        given direction u(x,y,z) of a seismic ray is equal to the        relative error r(x,y,z|m) defined by equation [9] or [10]; and    -   the perturbation induced by equations [11a] and [11b] may be        sufficiently coherent if the following inequality is satisfied:        −1<r(x,y,z|m)<3/2  [13]

A “coherent” perturbation may include any perturbation that meets one ormore predetermined criteria. These predetermined criteria may relate,for example, to fundamental physical laws or properties of the data. Forexample, predetermined criteria may include, entropy of the systemincreases as time increases, time functions monotonically increase,points on one side of a fault remain on the same side of the fault anddo not cross the fault as time changes, faults remain separate but maybranch, horizons remain separated by a continuous gap, the connectionsbetween faults and horizons are separated by no gap or a minimal gap.Alternatively or additionally, these predetermined criteria may relate,for example, to knowledge a user may have of the geophysical terrainbeing modeled. For example, the user may know (based on parallelcomputations or independent information) that a fault is either normalor reverse, branches, or crosses another fault. The user may add thisinformation to before or after data is processed for example as one ormore constraints on the data, as described herein.

A plurality of equiprobable coherent vector fields {v(x,y,z|m₁),v(x,y,z|m₂), . . . ,v(x,y,z|m_(n))} for perturbing models according tothe uncertainty induced by seismic velocity may be generated, forexample, by the following steps executed by a computing module (e.g.,executed by processor 140 described herein in reference to FIG. 10).Other operations or series of operations may be used, and the exact setof steps shown below may be varied.

-   -   1. Determine the unit vector field u(x,y,z) which may be        approximately tangential to a seismic ray that passes through        the corresponding point (x,y,z). Seismic ray tracing techniques        may be used to determine the vector field u(x,y,z). In another        embodiment, if the horizons are approximately horizontal, then        u(x,y,z) may be constant and equal to the unit vertical vector.    -   2. Cover a 3D geological domain of interest by a mesh M₂. Mesh        M₂ is preferably continuous across faults. M₂ may be a regular        mesh having substantially identically shaped (e.g., hexahedron        or tetrahedron) cells.    -   3. Select a random number (m) to be used as a “seed” by a random        function generator.    -   4. For each node of M₂, using a stochastic random function        generator, an occurrence r(x,y,z|m) may be numerically generated        satisfying equation [13]. For example, r(x,y,z|m) may be a        uniform random function with values in range [R₁,R₂] where R₁ is        greater than or equal to (−1) and R₂ is lesser than (3/2).    -   5. Create a new function e(x,y,z|m), such that:        -   a. e(x,y,z|m) is defined by sampling points at the nodes of            mesh M₂,        -   b. at any location (x,y,z), the value of e(x,y,z|m) can be            locally interpolated from the values sampled at the nodes of            mesh M₂ in the neighborhood of (x,y,z).    -   6. At each node of M₂ or on each cell of M₂, define e(x,y,z|m)        by equation [12], for example, by applying a constraint that        e(x,y,z|m) satisfies the equation, e.g., in a least squares        sense. For example, the DSI method or any other suitable method        may be used to approximate e(x,y,z|m) while taking this        constraint into account as directional gradient constraint.    -   7. For each sampling location (x_(s),y_(s),z_(s)), e.g., at well        markers, where the geometric error may be substantially        negligible, apply a control-point constraint, for example,        specifying that:        e(x _(s) ,y _(s) ,z _(s) |m)=null vector  [14]    -   8. If no control-point constraint is applied at step (7), then        choose (e.g., at least) one arbitrary minimum error point        (x_(s),y_(s),z_(s)) and apply a control-point constraint        according to equation [14].    -   9. Smoothness constraints are applied to specify that e(x,y,z|m)        should be as smooth as possible in any predefined specific sense        over the entire mesh M₂.    -   10. The DSI method or any other equivalent interpolation method        known in the art is then applied to compute the values of        e(x,y,z|m) at the vertices of mesh M₂ according to the        constraints above.    -   11. At each node location (x_(n),y_(n),z_(n)) of mesh M₂, use        equation [11b] to compute the value v(x_(n),y_(n),z_(n)|m) to        assign to that node, for example, as follows:        v(x _(n) ,y _(n) ,z _(n) |m)=e(x _(n) ,y _(n) ,z _(n)        |m)·u(x,y,z)  [15]    -   where the symbol “(e·u)” represents the product of a scalar (e)        by a vector (u).    -   12. If a subsequent equiprobable occurrence (m*) exists, return        to step (3) to generate a subsequent v(x,y,z|m*) with the        occurrence (m*).    -   13. Stop

In step (9) above, any “smoothness constraint” may be used, defined inthe neighborhood of each node (n) of mesh M₂, for example, by one ormore of the following (other constraints may be used):

-   -   Smoothness constraint #1: In the neighborhood of (n), the        gradient of e(x,y,z|m) is constrained to be as constant as        possible;    -   Smoothness constraint #2: In the neighborhood of (n), the        function e(x,y,z|m) is constrained to vary linearly, as much as        possible;    -   Smoothness constraint #3: In the neighborhood of (n), the        function e(x,y,z|m) is constrained to be as close as possible to        a harmonic function.

As described herein, the vector field v(x,y,z|m) may be athree-dimensional field that may be applied to the initial model to moveindividual nodes of the initial model in different directions in a 3Dspace.

In this embodiment, the magnitude of the displacement vector,e(x_(n),y_(n),z_(n)|m), is a measure of uncertainty, i.e., error, at theindividual node (e.g., as shown in equations [15]). Accordingly, themagnitude of the displacement of each node may correspond to (e.g., bebounded by) the error associated with that node. For example, nodesassociated with geological points that have relatively smalluncertainty, e.g., at well markers, are moved by a small or negligibledistance for all occurrences (m). Conversely, nodes associated withgeological points that have relatively high uncertainty are moved, forat least some occurrences of (m), by relatively large distances.Although each node may be moved by a different value for each occurrence(m), the error at each node defines the range of distances theindividual node may be moved for all occurrences of (m₁, m₂, . . . ).

In one embodiment, the direction of the displacement vector is in thedirection of the unit vectors u(x,y,z) (e.g., as shown in equations[15]). As described above in reference to equations [9] and [10], movingthe point (x,y,z) in direction u(x,y,z) may be equivalent to varying theerror in the seismic velocity S(x,y,z) of the node. Thus, applying athree-dimensional vector field of unit vector u(x,y,z), for example,associated with a function e(x,y,z|m), to perturb an initial model maymove an individual point in different directions in a 3D space accordingto the uncertainty of the error in the seismic velocity S(x,y,z) at thatpoint.

The uncertainty or error at each node may be stored in a memory (e.g.,memory 150 of FIG. 10) and retrieved by a processor (e.g., processor 140of FIG. 10). The processor may use the error at each node to calculatethe corresponding displacement vector, as described herein. Thedisplacement vector generated by the processor may be stored in thememory for later use.

Combining a Plurality of Perturbation Vector Fields

Perturbations by vector field are determined based on the uncertaintyrelated to a geological property of the model. In one example, thelocation of a fault in the model may be uncertain. As described in thesection entitled ‘Generating a Vector Field of Coherent PerturbationAssociated With a Fault,’ a vector field may be generated correspondingto the uncertainty in the location of the fault. In another example, theseismic velocity of model data is uncertain. As described in the sectionentitled ‘Generating a Vector Field of Coherent Perturbations Associatedwith Seismic Velocity,’ a vector field may be generated corresponding tothe uncertainty of the seismic velocity of the model data.

In one embodiment of the invention, multiple distinct vector fields,each corresponding to uncertainty of a distinct geophysical problem, maybe combined to generate a single vector field. The single combinedvector field may therefore correspond to the combined uncertainties ofmultiple distinct geophysical problems. When the combined vector fieldis applied to an initial model, the model is perturbed based on all thepossible solutions to multiple different geophysical problems.

A particle of sediment at a location (x,y,z) in an initial model may bemoved to a new location (x,y,z|m,s) for an occurrence (m) according toequation [7] duplicated here, as follows:(x,y,z|m,s)=(x,y,z)+v(x,y,z|m,s)  [7]

where an index (s) represents a specific source of uncertainty.

In geophysical data, there are typically many sources of uncertainties{s₁, s₂, . . . , s_(q)} that affect the geometry of the initial model.For example:

-   -   s₁ may correspond to the uncertainty associated to a fault F₁    -   s₂ may correspond to the uncertainty associated to a fault F₂    -   . . .    -   s_(q) may correspond to the uncertainty associated with seismic        velocity

To model the combined uncertainty associated with all the sources ofuncertainties {s₁, s₂, . . . ,s_(q)}, a combined vector field,v(x,y,z|m), is generated. The combined vector field, v(x,y,z|m), may be,for example, the sum of the individual vector fields associated witheach sources:v(x,y,z|m)=v(x,y,z|m,s1)+v(x,y,z|m,s2)+ . . . . +v(x,y,z|m,sq)  [16]

To generate a combined vector field that is coherent, for example, thefollowing steps may be executed by a computing module (other operationsor series of operations may be used, and the exact set of steps shownbelow may be varied):

-   -   1. Cover a 3D geological domain of interest by a mesh M₂. Mesh        M₂ is preferably continuous across faults. M₂ may be a regular        mesh having substantially identically shaped (e.g., hexahedron        or tetrahedron) cells.    -   2. Select a random number (m) to be used as a “seed” by a random        function generator.    -   3. Generate a series of equiprobable vector fields        {v(x,y,z|m,s₁), v(x,y,z|m,s₂), . . . , v(x,y,z|m,s_(q))}        modeling the occurrence (m) of uncertainties for each source of        uncertainty {s₁, s₂, . . . , s_(q)}, respectively, e.g., using a        stochastic simulator. For example the level of uncertainty of        the sources may be a measure of the extent to which each affects        the location of a point (x,y,z) corresponding to each particle        of sediment in the initial model. These vector fields may be        sampled at each node of M₂.    -   It may be appreciated that, for example:        -   a. It is preferred, but not required, that each vector field            v(x,y,z|m,s) satisfy equation [8].        -   b. Any stochastic simulator may be used to generate            v(x_(n),y_(n),z_(n)|m,s) at each node location            (x_(n),y_(n),z_(n)) of mesh M₂.    -   4. Create a new vector field v(x,y,z|m) such that:        -   a. v(x,y,z|m) is defined by sampling points at the nodes of            mesh M₂,        -   b. At any location (x,y,z), the value of v(x,y,z|m) may be            locally approximated from the values sampled at the nodes of            mesh M₂ in a neighborhood of (x,y,z).        -   c. v(x,y,z|m) is defined as the sum of the perturbation            fields {v(x,y,z|m,s1), v(x,y,z|m,s2), . . . ,            v(x,y,z|m,s_(q))}, e.g., according to equation [16].    -   5. Compute the divergence of v(x,y,z|m) at each node or on each        cell of the mesh M₂    -   6. If there is a node or a cell where the divergence is lower        than (−1) or greater than (3/2):        -   a. discard v(x,y,z|m), or        -   b. proceed to step (1) of the “Correcting Phase” of the            section entitled “Testing and Correcting Perturbations            Vector Fields” to correct v(x,y,z|m).    -   7. If a subsequent equiprobable occurrence v(x,y,z|m) may be        generated, return to step (2) above using the new occurrence.    -   8. Stop        Rapid Generation of Occurrences of a Perturbation Vector Field

A perturbation vector field v(x,y,z|m) corresponding to a given sourceof uncertainty may have multiple independent stochastic occurrences{v(x,y,z|m₁), v(x,y,z|m₂), . . . , v(x,y,z|m_(n))}, each of which affectthe location of particles of sediment corresponding to an initial modelof the subsurface. Assume that each of these independent occurrencessatisfies equation [8] substantially duplicated, as follows:−1<div(v(x,y,z|m _(i)))<3/2 for all i in {1, 2, . . . n}  [17]

A vector field w(x,y,z|m) may be defined to include blendingcoefficients B_(i)(m) (e.g., random variables), e.g., as follows:w(x,y,z|m)=B _(i)(m)·v(x,y,z|m ₁)+ . . . +Bn(n)·v(x,y,z|m _(n))  [18]

-   -   where the symbol “(B_(i)·v)” represents the product of a scalar        (B_(i)) by a vector (v).

The divergence operator is a linear operator. The blending coefficientsB_(i)(m) may be defined such that equations [19a] and [19b] aresatisfied:0≦B _(i)(m)≦1  [19a]B ₁(m)+B ₂(m)+ . . . +B _(n)(m)=1  [19b]

Evaluating equations [15], [19a] and [19b], the vector field w(x,y,z|m)satisfies the following inequality:−1<div(w(x,y,z|m))<3/2  [20]

As a consequence, w(x,y,z|m) may be defined by v(x,y,z|m). In practice,equation [18] provides a very fast way to generate stochasticoccurrences of v(x,y,z|m)=w(x,y,z|m).

To generate a series of coherent perturbation vector fields v(x,y,z|m),for example, the following steps may be executed by a computing module(other operations or series of operations may be used, and the exact setof steps shown below may be varied):

-   -   1) Cover a 3D geological domain of interest by a mesh M₂. Mesh        M₂ is preferably continuous across faults. M₂ may be a regular        mesh having substantially identically shaped (e.g., hexahedron        or tetrahedron) cells.    -   2) Generate a series of equiprobable vector fields {v(x,y,z|m₁),        v(x,y,z|m₂), . . . ,v(x,y,z|m_(n)} that satisfy equation [17],        for example, using a stochastic simulator.    -   3) Select a random number (m) to be used as a “seed” by a random        function generator.    -   4) Generate random blending coefficients {B₁(m), B₂(m), . . . ,        B_(n)(m)} that satisfy equations [19a] and [19b].    -   5) At each node (x_(n),y_(n),z_(n)) of mesh M₂, compute        v(x_(n),y_(n),z_(n)|m), for example, as follows:        v(x _(n) ,y _(n) ,z _(n) |m)=B ₁(m)·v(x _(n) ,y _(n) ,z _(n) |m        ₁)+ . . . +B _(n)(n)·v(x _(n) ,y _(n) ,z _(n) |m _(n))    -   where the symbol “(B_(i)·v)” represents the product of a scalar        (B_(i)) by a vector (v).    -   6) If a subsequent equiprobable occurrence may be generated,        return to step (3) above using the new occurrence.    -   7) Stop        Modeling Uncertainty Related to the Positioning of Particles of        Sediment (a Gradient Scalar Field Approach)

As previously described, a three-dimensional vector field v(x,y,z) maymove each node of an initial model in different directions in a 3Dspace. Accordingly, the three components v_(x), v_(y), and v_(z), ofv(x,y,z) are typically stored for each node. However, in anotherembodiment of the invention, a gradient of a scalar field f(x,y,z) maybe used instead of the three-dimensional vector field v(x,y,z). Thegradient of scalar field f(x,y,z) is a three-dimensional vector fieldgrad f(x,y,z) whose components are equal to the partial derivatives off(x,y,z) relative to x, y and z, respectively. Furthermore, the scalarfield f(x,y,z) may have only a single scalar value,f(x_(n),y_(n),z_(n)), stored for each node (n) whose coordinates are(x_(n),y_(n),z_(n)). Therefore, the gradient of a scalar field f(x,y,z)may be used to generate three-dimensional vector field perturbations ofthe initial model while significantly reducing the effective amount ofdata stored for the vector fields (e.g., by ⅔).

Accordingly less computational resources may be used when the vectorfield v(x,y,z|m) is derived from a scalar field f(x,y,z|m), for example,as follows:v(x,y,z|m)=grad f(x,y,z|m)  [21]

where {grad f} represents the gradient of f(x,y,z|m)

It is well known that the divergence of v(x,y,z|m) is equal to theLaplacian Δf(x,y,z|m) of f(x,y,z|m):div(v(x,y,z|m))=Δf(x,y,z|m)  [22]

Therefore, for the gradient of the scalar field f(x,y,z) to honor theinequality [8], the function f(x,y,z) may be built in such a way thatits Laplacian belongs to the range [−1,3/2] at any location in thestudied domain:−1<Δf(x,y,z|m)<3/2  [23]

A plurality of scalar fields {f(x,y,z|m₁), f(x,y,z|m₂, . . . ,f(x,y,z|m_(n)} may be generated that satisfy equation [23]. These scalarfields may be tested and corrected to ensure that when thethree-dimensional gradients of the scalar fields are applied to aninitial model, the plurality of perturbed models generated are coherent,for example, by a computing module executing the following steps:

Testing and Correcting a Perturbations Scalar Field

A plurality of scalar fields {f(x,y,z|m₁), f(x,y,z|m₂, . . . ,f(x,y,z|m_(n)} may be tested and corrected to ensure the aforementionedcoherence constraints are satisfied, for example, by a computing moduleexecuting the following steps (other operations or series of operationsmay be used, and the exact set of steps shown below may be varied):

Testing Phase

-   -   1. Cover a 3D geological domain of interest by a mesh M₂. Mesh        M₂ is preferably continuous across faults. M₂ may be a regular        mesh having substantially identically shaped (e.g., hexahedron        or tetrahedron) cells.    -   2. Select a random number (m) to be used as a “seed” by a random        function generator.    -   3. Use a stochastic generator to generate an equiprobable        occurrence f(x,y,z|m) of a perturbation scalar field.    -   4. Compute the Laplacian of f(x,y,z|m) at each node or on each        cell of mesh M₂ and determine:        -   a. If the Laplacian is greater than (−1) and lower than            (3/2) everywhere, then proceed to step (5)        -   b. Otherwise proceed according to one of the following:            -   i. discard f(x,y,z|m), or            -   ii. proceed to step (1) of the “Correcting Phase” of                this section to correct f(x,y,z|m).    -   5. If a subsequent equiprobable occurrence f(x,y,z|m) may be        generated, return to step (2) above using the new occurrence    -   6. Stop        Correcting Phase

When a perturbation scalar field f(x,y,z|m) is determined to violateequation [23], as in step 4b(ii) above, the scalar field f(x,y,z|m) maybe corrected to satisfy equation [23], for example, by the followingsteps executed by a computing module (other operations or series ofoperations may be used, and the exact set of steps shown below may bevaried):

-   -   1. Accept a scalar field f(x,y,z|m) determined to violate        equation [23].    -   2. Create a copy f₀(x,y,z|m) of the scalar field f(x,y,z|m)    -   3. For each node (n) of mesh M₂, determine the coordinate        location (x_(n),y_(n),z_(n)) and apply a constraint specifying        that a new value of f(x_(n),y_(n),z_(n)|m) approximated at this        location may be approximately equal to f₀(x_(n),y_(n),z_(n)|m),        for example, in a least squares sense. For example, a “soft” (or        “fuzzy”) control node may be used.    -   4. At each node of M₂ or on each cell of M₂, apply a constraint        specifying that new values of f(x,y,z|m) satisfy equation [23].        For example, the DSI method or any other equivalent        approximation method may be used to approximate f(x,y,z|m) with        equation [23] taken into account as an inequality constraint.    -   5. A smoothness constraint may be applied specifying that the        new values of f(x,y,z|m) be as smooth as possible in the modeled        domain.    -   6. For example, the DSI method or any other equivalent        approximation method may be used to compute the new values of        f(x,y,z|m) at the nodes of mesh M₂ while satisfying the equation        [23] and any additional constraints.    -   7. Stop

In step (5) above, a “smoothness constraint” may be used, defined in theneighborhood of each node (n) of mesh M₂, for example, by one or more ofthe following (other smoothness constraints may be used):

-   -   Smoothness constraint #1: In the neighborhood of (n), the        gradient of f(x,y,z|m) is constrained to be as constant as        possible;    -   Smoothness constraint #2: In the neighborhood of (n), the        function f(x,y,z|m) is constrained to vary linearly, as much as        possible;    -   Smoothness constraint #3: In the neighborhood of (n), the        function f(x,y,z|m) is constrained to be as close as possible to        a harmonic function.        Generating a Gradient Scalar Field of Coherent Perturbations        Associated with a Fault

Faults are generally observed and sampled on seismic cross sections orautomatically extracted from seismic cubes. Since faults often appear“fuzzy” on seismic cross sections, the exact location of faults may beuncertain and thus, it may be difficult to position faults at an exactlocation on a 3D mesh M₁ of a geological model.

According to embodiments of the invention, when the position of a faultis uncertain gradients of a plurality of scalar fields may be generated,each of which may be applied to an initial model of the fault, which inturn generates a plurality of perturbed fault models. The appliedgradient scalar fields may be generated for example, by the followingsteps executed by a computing module (other operations or series ofoperations may be used, and the exact set of steps shown below may bevaried).

-   -   1. Cover a 3D geological domain of interest by a mesh M₂. Mesh        M₂ is preferably continuous across faults. M₂ may be a regular        mesh having substantially identically shaped (e.g., hexahedron        or tetrahedron) cells.    -   2. Select a random number (m) to be used as a “seed” by a random        function generator.    -   3. Generate a new scalar field f(x,y,z|m) to be built in the 3D        studied domain for each equiprobable stochastic occurrence (m)        such that:        -   a. f(x,y,z|m) is defined by sampling values at the nodes of            mesh M₂,        -   b. at any location (x,y,z) in the 3D studied domain, the            value of f(x,y,z|m) may be locally approximated from the            values sampled at the nodes of mesh M₂ in the neighborhood            of (x,y,z).    -   4. Generate an equiprobable occurrence v_(F)(x,y,z|m) of a        vector field modeling the equiprobable stochastic occurrence (m)        of the uncertainty at point (x,y,z) defined on the surface of        fault F. The equiprobable occurrence v_(F)(x,y,z|m) of a vector        field may be generated using for example a stochastic simulator.        The vector field v_(F)(x,y,z|m) may be generated at each of a        set of given sampling points S_(F)={(x₁,y₁,z₁), . . . ,        (x_(q),y_(q),z_(q))} located on fault F. Any suitable stochastic        simulator or processing module may be used to generate the        equiprobable occurrence v_(F)(x,y,z|m).    -   5. For each sampling location (x_(f),y_(f),z_(f)) in set S_(F),        apply a constraint specifying that the value of the gradient of        f(x_(f),y_(f),z_(f)|m) approximated at location        (x_(f),y_(f),z_(f)) may be approximately equal to        v_(F)(x_(f),y_(f),z_(f)|m), for example in a least squares        sense. For example, a “soft” (or “fuzzy”) control node/point may        be used.    -   6. For each node (n) of M₂ having a coordinate location        (x_(n),y_(n),z_(n)) greater than a predetermined distance from        fault F, apply a constraint specifying that the gradient of        f(x_(n),y_(n),z_(n)|m) approximated at location        (x_(n),y_(n),z_(n)) may be approximately equal to a null vector,        for example in a least squares sense. For example, a “soft” (or        “fuzzy”) control node/point may be used.    -   7. If F is a secondary fault which branches from a main fault        F_(main) along a common border B, then:        -   a. Build a set S_(B) of sampling points close to B; and        -   b. For each point b of S_(B) whose coordinates are            (x_(b),y_(b),z_(b)), apply a constraint specifying that the            dot product of the gradient of f(x_(b),y_(b),z_(b)|m) with            the normal vector of F_(main) at location            (x_(b),y_(b),z_(b)) vanishes. Accordingly, the secondary            faults are perturbed in a direction of the main fault            F_(main) and move along the main fault F_(main) without            altering the general fault shape. Such a constraint is            linear and may be taken into account by the DSI method or an            equivalent approximation method.    -   8. At each node of M₂ or on each cell of M₂, apply on f(x,y,z|m)        a constraint specifying that inequality [23] is satisfied. For        example, if the DSI method is used to approximate f(x,y,z|m),        then the constraint may be expressed as two inequality        constraints.    -   9. Smoothness constraints may be applied to specify that        f(x,y,z|m) should be as smooth as possible in the neighborhood        of each node (n) of the mesh M₂.    -   10. For an arbitrary point (x₀,y₀,z₀) within the studied domain,        apply a control-point constraint specifying that f(x,y,z|m) may        be equal to a given arbitrary value f₀ at the location        (x₀,y₀,z₀). For example (x₀,y₀,z₀) may be a point close to the        center of the model and f₀ may be a null value.    -   11. For example, the DSI method or any other equivalent        approximation method may be used to compute the new values of        f(x,y,z|m) at the nodes of mesh M₂ while satisfying the        constraints above.    -   12. If a subsequent equiprobable occurrence (m*) exists, return        to step (2) to generate a subsequent f(x,y,z|m*) with the        occurrence (m*).    -   13. Stop

In step (9) above, a “smoothness constraint” may be used, defined in theneighborhood of each node (n) of mesh M₂, for example, by one or more ofthe following (other constraints may be used):

-   -   Smoothness constraint #1: In the neighborhood of (n), the        gradient of f(x,y,z|m) is constrained to be as constant as        possible;    -   Smoothness constraint #2: In the neighborhood of (n), the        function f(x,y,z|m) is constrained to vary linearly, as much as        possible;    -   Smoothness constraint #3: In the neighborhood of (n), the        function f(x,y,z|m) is constrained to be as close as possible to        a harmonic function.        Combining a Plurality of Perturbation Scalar Fields

Perturbations by gradient scalar fields are determined based on theuncertainty related to a geological property of the model. Similarly tothe description in the section entitled “Combining a Plurality ofPerturbation Vector Fields,” multiple distinct gradient scalar fields,each corresponding to uncertainty of a distinct geophysical problem, maybe combined to generate a single gradient scalar field. The singlecombined gradient scalar field may therefore correspond to the combineduncertainties of multiple distinct geophysical problems. When thecombined gradient scalar field is applied to an initial model, the modelis perturbed based on all the possible solutions to multiple differentgeophysical problems.

A particle of sediment at a location (x,y,z) in an initial model may bemoved to a new location (x,y,z|m,s) for an occurrence (m) according toequation [7]. In geophysical data, there are typically many sources ofuncertainties {s₁, s₂, . . . , s_(q)} that affect the geometry of theinitial model. For example:

-   -   s_(i) may correspond to the uncertainty associated to a fault F₁    -   s₂ may correspond to the uncertainty associated to a fault F₂    -   . . .    -   s_(q) may correspond to the uncertainty associated with seismic        velocity

To model the combined uncertainty associated with all the sources ofuncertainties {s₁, s₂, . . . , s_(q)}, a combined scalar fieldf(x,y,z|m) may be generated. The combined scalar field, f(x,y,z|m), maybe, for example, the sum of the individual vector fields associated witheach source:f(x,y,z|m)=f(x,y,z|m,s _(i))+f(x,y,z|m,s ₂)+ . . . . +f(x,y,z|m,s_(q))  [24]

To generate a combined vector field that is coherent, for example, thefollowing steps may be executed by a computing module (other operationsor series of operations may be used, and the exact set of steps shownbelow may be varied):

-   -   1. Cover a 3D geological domain of interest by a mesh M₂. Mesh        M₂ is preferably continuous across faults. M₂ may be a regular        mesh having substantially identically shaped (e.g., hexahedron        or tetrahedron) cells.    -   2. Select a random number (m) to be used as a “seed” by a random        function generator.    -   3. Generate a series of equiprobable scalar fields        {f(x,y,z|m,s_(i)), f(x,y,z|m,s₂), . . . , f(x,y,z|m,s_(q))}        having gradients that model the occurrence (m) of uncertainties        for each source of uncertainty {s₁, s₂, . . . , s_(q)},        respectively, e.g., using a stochastic simulator. For example        the level of uncertainty of the sources may be a measure of the        extent to which each affects the location of a point (x,y,z)        corresponding to each particle of sediment in the initial model.        These scalar fields may be sampled at each node of M₂.    -   It may be appreciated that, for example:        -   a. It is preferred, but not required, that each scalar field            f(x,y,z|m,s) satisfy equation [23].        -   b. Any stochastic simulator may be used to generate            f(x_(n),y_(n),z_(n)|m,s) at each node location            (x_(n),y_(n),z_(n)) of mesh M₂.    -   4. Create a new scalar field f(x,y,z|m) such that:        -   a. The scalar field f(x,y,z|m) is defined by sampling points            at the nodes of mesh M₂,        -   b. At any location (x,y,z), the value of f(x,y,z|m) may be            locally approximated from the values sampled at the nodes of            mesh M₂ in a neighborhood of (x,y,z).        -   c. The scalar field f(x,y,z|m) is defined as the sum of the            perturbation fields {f(x,y,z|m,s1), f(x,y,z|m,s2), . . . ,            f(x,y,z|m,sq)}, e.g., according to equation [24].    -   5. Compute the Laplacian of f(x,y,z|m) at each node or on each        cell of the mesh M₂.    -   6. If there is a node or a cell where the Laplacian is lower        than (−1) or greater than (3/2):        -   a. discard f(x,y,z|m), or        -   b. Or proceed to step (1) of the “Correcting Phase” of the            section entitled “Testing and Correcting Perturbations            Scalar Fields” to correct f(x,y,z|m).    -   7. If a new equiprobable coherent occurrence of f(x,y,z|m) may        be generated, return to step (2) above using the new occurrence.    -   8. Stop        Rapid Generation of Occurrences of a Perturbation Scalar Field

A scalar field f(x,y,z|m) may have multiple independent stochasticoccurrences {f(x,y,z|m₁), f(x,y,z|m₂), f(x,y,z|m_(n))}, each of whichmay affect the modeled location of particles of sediment correspondingto an initial model of the subsurface, for example, according toequations [7] and [21]. Assume that each of these independentoccurrences honors the coherency equation [23] substantially duplicated,as follows:−1<Δf(x,y,z|m _(i))<3/2 for all i in {1, 2, . . . ,n}  [25]

A scalar field F(x,y,z|m) may be defined to include blendingcoefficients B_(i)(m) (e.g., random variables), e.g., as follows:F(x,y,z|m)=B ₁(m)·f(x,y,z|m ₁)+ . . . +B _(n)(n)·f(x,y,z|m _(n))  [26]

-   -   where the symbol “(B_(i)·f)” represents the product of a scalar        (B_(i)) by a vector (f).

It is well known that the Laplacian operator is a linear operator.Therefore, if the blending coefficients B_(i)(m) are defined such thatequations [19a] and [19b] are satisfied (e.g., as described in pages519-525 of [MAL02]):0≦B _(i)(m)≦1  [19a]B _(i)(m)+B ₂(m)+ . . . +B _(n)(m)=1  [19b]

then the scalar field F(x,y,z|m) satisfies the following coherencyinequality:−1<div(F(x,y,z|m))<3/2  [27]

As a consequence, F(x,y,z|m) may be used as a stochastic occurrencef(x,y,z|m) of the scalar field. In practice, equation [26] provides avery fast way to generate stochastic occurrences off(x,y,z|m)=F(x,y,z|m) and, as a consequence, stochastic occurrencesv(x,y,z|m) may be rapidly generated as gradients of F(x,y,z|m).

To generate a series of coherent perturbation scalar fields f(x,y,z|m),for example, the following steps may be executed by a computing module(other operations or series of operations may be used, and the exact setof steps shown below may be varied):

-   -   1) Cover a 3D geological domain of interest by a mesh M₂. Mesh        M₂ is preferably continuous across faults. M₂ may be a regular        mesh having substantially identically shaped (e.g., hexahedron        or tetrahedron) cells.    -   2) Generate a series of equiprobable vector fields {f(x,y,z|m₁),        f(x,y,z|m₂), . . . , f(x,y,z|m_(n)} that satisfy equation [25],        for example, using a stochastic simulator.    -   3) Select a random number (m) to be used as a “seed” by a random        function generator.    -   4) Generate random blending coefficients {B₁(m), B₂(m), . . . ,        B_(n)(m)} that satisfy equations [19a] and [19b].    -   5) At each node (x_(n),y_(n),z_(n)) of mesh M₂, compute        F(x_(n),y_(n),z_(n)|m), for example, as follows:        f(x _(n) ,t _(n) ,z _(n) |m)=B ₁(m)·f(x _(n) ,y _(n) ,z _(n) |m        ₁)+ . . . +B _(n)(n)·f(x _(n) ,y _(n) ,z _(n) |m _(n))    -   where the symbol “(B_(i)·f)” represents the product of a scalar        (B_(i)) by a vector (v).    -   6) If a subsequent equiprobable occurrence may be generated,        return to step (3) above using the new occurrence.    -   7) Stop        Modeling and Visualizing Global Uncertainty Related to the        Geometry of the Subsurface        Modeling Uncertainty on Faults and Horizons Jointly

In the description above, for each stochastic occurrence (m), horizonsand faults may be modeled separately. To generate horizons and faultssimultaneously or together, for example, the following steps may beexecuted by a computing module (other operations or series of operationsmay be used, and the exact set of steps shown below may be varied):

-   -   1. Generate a plurality of geological-time functions        {t(x,y,z|m_(t1)), t(x,y,z|m₂), . . . , t(x,y,z|m_(tn))}, each of        which may be defined at nodes of a given 3D mesh M₁. For        example, the plurality of geological-time functions may be        generated according to the section entitled “Modeling        Uncertainties on Horizons.”    -   2. Generate a series of equiprobable stochastic coherent vector        fields {v(x,y,z|m_(v1)), v(x,y,z|m_(v2)), . . . ,        v(x,y,z|m_(vk))}. For example, the vector fields may be        generated according to the sections entitled “Testing and        Correcting Perturbations Vector Fields” and “Rapid Generation of        Occurrences of a Perturbation Scalar Field.”    -   3. For each stochastic occurrence (m_(t)) in the sequence        {m_(t1), m_(t2), . . . , m_(tn)}:        -   a. Randomly select a stochastic occurrence (m_(v)) in the            sequence {m_(v1), m_(v2), . . . , m_(vk)}        -   b. Build a copy M₁ (m_(v)) of M₁        -   c. Move each node (n) of M₁(m_(v)) to a new location            (x_(n),y_(n),z_(n)|m_(v)) of (n) derived from the initial            location (x_(n),y_(n),z_(n)) as follows:            (x _(n) ,y _(n) ,z _(n) |m _(v))=(x _(n) ,y _(n) ,z            _(n))+v(x _(n) ,y _(n) ,z _(n) |m _(v))        -   d. Set the value of each node (n) of M₁(m_(v)) with            coordinates, (x_(n),y_(n),z_(n)), to be the values of            sampling points of the geological-time function            t(x_(n),y_(n),z_(n)|m_(t)) previously computed.    -   4. Stop

For each pair (m_(t),m_(v)), the nodes (n) of mesh M₁(m_(v)) have arandom location and a random occurrence of the geological-timet(x,y,z|m_(t)). Accordingly, each pair {M₁(m_(v)), t(x,y,z|m_(t))} maybe used as a coherent stochastic model of both horizons and the faults.

Reference is made to FIG. 8, which schematically illustrates a displayof a plurality of equiprobable perturbed models GM(m₁) 802, GM(m₂) 804,GM(m₃) 806 generated from an initial model 800, according to anembodiment of the invention. The plurality of equiprobable perturbedmodels were generated according to embodiments of the process describedin this section.

Modeling the Probability of Leakage Through Faults

Reference is made to FIGS. 9A and 9B, which schematically illustratefaults 900 across which hydrocarbon deposits in a reservoir 904 may notleak and may leak, respectively, from one side of the fault 906 to theother side of the fault 908 according to an embodiment of the invention.Depending on the location of top and bottom horizons of a reservoir 904,the hydrocarbons located on one side of a fault F 906 may leak to theother side of F 908. Different occurrences of the equiprobable models,e.g., GM(m₁), GM(m₂), GM(m₃), may be used to identify the probability ofsuch leakage. Determining the probability of the leakage of hydrocarbonsthrough a fault F 900 may inform reservoir engineers as to where tosearch for hydrocarbon sources, for example, as follows.

-   -   If the probability of leakage is low, then two wells may be        drilled, one on each side of the fault F; and    -   if the probability of leakage is high, then a single well may be        drilled on one side of the fault F (e.g., eliminating the cost        of building a second well on a second side of the fault F).

To model the probability of leakage, p(n), through a modeled fault F900, for example, the following steps may be executed by a computingmodule, according to one embodiment of the invention (other operationsor series of operations may be used, and the exact set of steps shownbelow may be varied):

-   -   1) Identify two sides of a fault F 900, side F₀ 906 and side F₁        908.    -   2) Cover the fault F 900 with a fine 2D mesh M_(F).    -   3) At each node (n) of mesh M_(F) associated with the modeled        fault F 900, attach a counter p(n) initialized to 0.    -   4) For each stochastic occurrence GM(m) of an equiprobable        perturbation of the geometric model, for each node (n) of M_(F):        -   a. Consider the two points (n₀) 910 and (n₁) 912 collocated            with fault node (n) and located on the sides F₀ 906 and F₁            908 of fault F 900, respectively.        -   b. If points (n₀) 910 and (n₁) 912 are both located in            reservoir 904, then increment counter p(n) by one such that,            for example, p(n)=p(n)+1    -   5) For each node (n) of M_(F), divide p(n) by the total number        of stochastic realizations GM(m)    -   6) Stop

The values p(n) may represent an estimate of the probability of leakagethrough fault F 900 at each node (n) of M_(F). These values may bestored, for example, to be used by or displayed to a user. The valuesmay be displayed as a color scale overlaying a model. The color scalemay be normalized. Different probabilities or ranges of probabilitiesmay correspond to different colors. Accordingly, a user may inspect thedisplay to determine where there are relatively high or lowprobabilities of leakage through a fault F.

Visualizing the Probability of Leakage Through Faults

The probability of leakage p(n) determined above may be modeled as a 2Dor 3D visualization for viewing on a user display. To generate avisualization of the probability p(n) of leakage through a fault F, forexample, the following steps may be executed by a computing module(other operations or series of operations may be used, and the exact setof steps shown below may be varied):

-   -   1) Generate a 2D mesh M_(F) covering the fault F such that each        node (n) of mesh M_(F) contains the value p(n) of the        probability of leakage at the location of the node.    -   2) Using any appropriating graphing device (e.g., computer        system 130 as described in FIG. 10), generate a (e.g., 2D or 3D)        display with an appropriate visualization technique to view the        variations of p(n) on F. For example, visualization technique        may include:        -   drawing contour lines of p(n) onto F; and/or        -   painting F with colors such that each color corresponds to a            given value of p(n).    -   3) Stop

In one embodiment, the model displaying the probability of leakagethrough F may be displayed together with the spatial model GM(m). Forexample, a model displaying the probability of leakage through F may beoverlaid on, or adjacent to the spatial model GM(m), on a display suchas display 180 of FIG. 10). Alternatively, the probability of leakagethrough F may be displayed separately from the spatial model GM(m).Other arrangements or combinations of the aforementioned models may beused.

Reference is made to FIG. 10, which schematically illustrates a systemincluding a transmitter, receiver and computing system in accordancewith an embodiment of the present invention. Methods disclosed hereinmay be performed using a system 105 of FIG. 10.

System 105 may include a transmitter 190, a receiver 120, a computingsystem 130, and a display 180. The aforementioned data, e.g., seismicdata used to form intermediate data and finally to model subsurfaceregions, may be ascertained by processing data generated by transmitter190 and received by receiver 120. Intermediate data may be stored inmemory 150 or other storage units. The aforementioned processesdescribed herein may be performed by software 160 being executed byprocessor 140 manipulating the data.

Transmitter 190 may transmit signals, for example, acoustic waves,compression waves or other energy rays or waves, that may travel throughsubsurface (e.g., below land or sea level) structures. The transmittedsignals may become incident signals that are incident to subsurfacestructures. The incident signals may reflect at various transition zonesor geological discontinuities throughout the subsurface structures. Thereflected signals may include seismic data.

Receiver 120 may accept reflected signal(s) that correspond or relate toincident signals, sent by transmitter 190. Transmitter 190 may transmitoutput signals. The output of the seismic signals by transmitter 190 maybe controlled by a computing system, e.g., computing system 130 oranother computing system separate from or internal to transmitter 190.An instruction or command in a computing system may cause transmitter190 to transmit output signals. The instruction may include directionsfor signal properties of the transmitted output signals (e.g., such aswavelength and intensity). The instruction to control the output of theseismic signals may be programmed in an external device or program, forexample, a computing system, or into transmitter 190 itself.

Computing system 130 may include, for example, any suitable processingsystem, computing system, computing device, processing device, computer,processor, or the like, and may be implemented using any suitablecombination of hardware and/or software. Computing system 130 mayinclude for example one or more processor(s) 140, memory 150 andsoftware 160. Data 155 generated by reflected signals, received byreceiver 120, may be transferred, for example, to computing system 130.The data may be stored in the receiver 120 as for example digitalinformation and transferred to computing system 130 by uploading,copying or transmitting the digital information. Processor 140 maycommunicate with computing system 130 via wired or wireless command andexecution signals.

Memory 150 may include cache memory, long term memory such as a harddrive, and/or external memory, for example, including random accessmemory (RAM), read only memory (ROM), dynamic RAM (DRAM), synchronousDRAM (SD-RAM), flash memory, volatile memory, non-volatile memory, cachememory, buffer, short term memory unit, long term memory unit, or othersuitable memory units or storage units. Memory 150 may storeinstructions (e.g., software 160) and data 155 to execute embodiments ofthe aforementioned methods, steps and functionality (e.g., in long termmemory, such as a hard drive). Data 155 may include, for example, rawseismic data collected by receiver 120, instructions for building amesh, instructions for partitioning a mesh, and instructions forprocessing the collected data to generate a model, or other instructionsor data. Memory 150 may also store instructions to remove local maximaand minima points (“bubbles”) from horizons of a model. Memory 150 maystore a geological-time function, a model representing a structure whenit was originally deposited (e.g., in uvt-space) and/or a modelrepresenting the corresponding structure in a current time period (e.g.,in xyz-space). Memory 150 may store cells, nodes, voxels, etc.,associated with the model and the model mesh. Memory 150 may also storeforward and/or reverse uvt-transformations to transform current modelsin xyz-space to models in uvt-space, and vice versa. Memory 150 maystore instructions to perturb an initial model and data associated withthe plurality of perturbed models generated therefrom. Memory 150 mayalso store the three-dimensional vector fields (or one-dimensionalscalar fields), which when applied to the nodes of the initial model (orwhen the gradient of the scalar field is applied to the nodes of theinitial model), move the nodes of the initial model to generate one ofthe plurality of perturbed models. Applying a vector field tocorresponding nodes of the initial model may cause the nodes to “move”,“slide”, or “rotate”, thereby transforming modeled geological featuresrepresented by nodes and cells of the initial model. Data 155 may alsoinclude intermediate data generated by these processes and data to bevisualized, such as data representing graphical models to be displayedto a user. Memory 150 may store intermediate data. System 130 mayinclude cache memory which may include data duplicating original valuesstored. For example, memory 150 may store the set of erroneous orpathological nodes, P, (e.g., local maxima and minima points) which areremoved from the horizons of models. Memory 150 may also store measuresof error or uncertainty at the nodes of an initial model. Memory 150 maystore measures of the divergence of vector fields to determine if thedivergences are within a predetermined threshold according to equations[8] and [23]. Cache memory may include data duplicating original valuesstored elsewhere or computed earlier, where the original data may berelatively more expensive to fetch (e.g., due to longer access time) orto compute, compared to the cost of reading the cache memory. Cachememory may include pages, memory lines, or other suitable structures.Additional or other suitable memory may be used.

Computing system 130 may include a computing module havingmachine-executable instructions. The instructions may include, forexample, a data processing mechanism (including, for example,embodiments of methods described herein) and a modeling mechanism. Theseinstructions may be used to cause processor 140 using associatedsoftware 160 modules programmed with the instructions to perform theoperations described. Alternatively, the operations may be performed byspecific hardware that may contain hardwired logic for performing theoperations, or by any combination of programmed computer components andcustom hardware components.

Embodiments of the invention may include an article such as a computeror processor readable medium, or a computer or processor storage medium,such as for example a memory, a disk drive, or a USB flash memory,encoding, including or storing instructions, e.g., computer-executableinstructions, which when executed by a processor or controller, carryout methods disclosed herein.

Display 180 may display data from transmitter 190, receiver 120, orcomputing system 130 or any other suitable systems, devices, orprograms, for example, an imaging program or a transmitter or receivertracking device. Display 180 may include one or more inputs or outputsfor displaying data from multiple data sources or to multiple displays.For example display 180 may display visualizations of subsurface modelsincluding subsurface features, such as faults, horizons andunconformities. Display 180 may display an initial model as well as aplurality of models that are equiprobable perturbations of the initialmodel. The initial model and/or any of the plurality of perturbed modelsmay be displayed one at a time, two at a time, or many at a time (e.g.,the number selected by a user or automatically based on the differencebetween models or the total number of perturbed models). Display 180 maydisplay the models in a sequence of adjacent models, through which auser may scan (e.g., by clicking a ‘next’ or ‘previous’ button with apointing device such as a mouse or by scrolling through the models).

Input device(s) 165 may include a keyboard, pointing device (e.g.,mouse, trackball, pen, touch screen), or cursor direction keys, forcommunicating information and command selections to processor 140. Inputdevice 165 may communicate user direction information and commandselections to the processor 140. For example, a user may use inputdevice 165 to select one or more preferred models from among theplurality of perturbed models or to delete a non-preferred model (e.g.,by pointing a ‘select’ or ‘delete’ button on a display 180 monitoradjacent to the model using a cursor controlled by a mouse or byhighlighting and pressing a forward or delete key on a keyboard).

Processor 140 may include, for example, one or more processors,controllers or central processing units (“CPUs”). Software 160 may bestored, for example, in memory 150. Software 160 may include anysuitable software, for example, DSI software.

Processor 140 may generate an initial model, for example, using data 155from memory 150. A model may be generated without local maxima or minimaon surfaces of horizons. In one embodiment, a model may simulatestructural, spatial or geological properties of a subsurface region,such as, porosity or permeability through geological terrains.

Processor 140 may initially generate a three dimensional mesh, lattice,or collection of nodes that spans or covers a domain of interest. Thedomain may cover a portion or entirety of the three-dimensional spacesought to be modeled. Processor 140 may automatically compute the domainto be modeled and the corresponding mesh based on the collected seismicdata so that the mesh covers a portion or the entirety of thethree-dimensional space from which geological data is collected (e.g.,the studied subsurface region). Alternatively or additionally, thedomain or mesh may be selected or modified by a user, for example,entering coordinates or highlighting regions of a simulated optionaldomain or mesh. For example, the user may select a domain or mesh tomodel a region of the Earth that is greater than a user-selectedsubsurface distance (e.g., 100 meters) below the Earth's surface, adomain that occurs relative to geological features (e.g., to one side ofa known fault or riverbed), or a domain that occurs relative to modeledstructures (e.g., between modeled horizons H(t₁) and H(t₁₀₀)). Processor140 may execute software 160 to partition the mesh or domain into aplurality of three-dimensional (3D) cells, columns, or other modeleddata (e.g., represented by voxels, pixels, data points, bits and bytes,computer code or functions stored in memory 150). The cells or voxelsmay have hexahedral, tetrahedral, or any other polygonal shapes, andpreferably three-dimensional shapes. Alternatively, data may includezero-dimensional nodes, one-dimensional segments, two-dimensional facetand three-dimensional elements of volume, staggered in athree-dimensional space to form three-dimensional data structures, suchas cells, columns or voxels. The cells preferably conform to andapproximate the orientation of faults and unconformities. Each cell mayinclude faces, edges and/or vertices. Each cell or node may correspondto one or more particles of sediment in the Earth (e.g., a node mayinclude many cubic meters of earth, and thus many particles). As shownin FIG. 1, model 100 in G*-space 106, i.e., modeling seismic data as afunction of geological time (the time the structures being modeled wereoriginally deposited) may be generated using a regular mesh (e.g.,having a uniform lattice or grid structure or nodes and cells). Incontrast, model 108 of FIG. 1 may be generated using an irregular mesh(having an irregular lattice or arrangement or nodes or cells). Datacollected by receiver 120 after the time of deposition in a current orpresent time period, faults and unconformities that have developed sincethe original time of deposition, e.g., based on tectonic motion,erosion, or other environmental factors, may disrupt the regularstructure of the geological domain. Accordingly, an irregular mesh maybe used to model current geological structures, for example, so that atleast some faces, edges, or surfaces of cells are oriented parallel tofaults and unconformities, and are not intersected thereby. In oneembodiment, a mesh may be generated based on data collected by receiver120, alternatively, a generic mesh may be generated to span the domainand the data collected by receiver 120 may be used to modify thestructure thereof. For example, the data collected may be used togenerate a set of point values at “sampling point”. The values at thesepoints may reorient the nodes or cells of the mesh to generate a modelthat spatially or otherwise represents the geological data collectedfrom the Earth. Other or different structures, data points, or sequencesof steps may be used to process collected geological data to generate amodel. The various processes described herein (e.g., modeling faults)may be performed by manipulating such modeling data.

The geological time function may be defined at a finite number of nodesor sampling points based on real data corresponding to a subsurfacestructure, e.g., one or more particles or a volume of particles ofEarth. The geological time function may be approximated between nodes tocontinuously represent the subsurface structure, or alternatively,depending on the resolution in which the data is modeled may representdiscrete or periodic subsurface structures, e.g., particles or volumesof Earth that are spaced from each other.

Seismic data used to generate the model may be uncertain. For example,the wavelength of a signal received by receiver 120 which was reflectedoff of a seismic terrain may have an uncertainty proportional to theincident wavelength of the signal. When the seismic data is uncertain,the computing system 130 may generate a geophysical model that is alsouncertain. That is, the exact solution may be indeterminate at one ormore nodes of the geophysical model. Instead, there is a plurality ofpossible solutions, each with the same probability of being correct.

Embodiments of the invention propose a solution to overcome theuncertainty by perturbing an initial uncertain model to generate anddisplay a plurality of perturbed models, each perturbed model having adifferent arrangement of nodes, where each arrangement has substantiallythe same probability of occurring in the subsurface terrain. In analternate embodiment, the arrangement of nodes in the perturbed modelsmay have approximately the same probability, for example, within apredetermined range of probabilities. Additionally or alternatively, thearrangement of nodes may have approximately the same probability ofoccurring with respect to a first geological property, for example, thelocation of faults, and a different probability with respect to a secondgeological property, for example, the density or porosity of theterrain.

Computing system 130 may generate each perturbed model by for exampleapplying a vector field (e.g., part of data 155) to the initial model(e.g., by vector addition). The applied vector field typically causesthe nodes of the initial model to move to corresponding new locations inthe perturbed models. The vector fields are preferably substantiallycontinuous. A plurality of such vector fields may be individuallyapplied to the initial model, thereby generating a plurality ofdifferent models, each of which has substantially the same probabilityof occurrence.

According to embodiments of the invention, the vector fields applied tothe initial model are three-dimensional. That is, the vectors of thevector field may have different directions represented (non-trivially)by a three-component vector (v_(x), v_(y), v_(z)). Accordingly, when thethree-dimensional vector field is applied to the initial model, each ofthe nodes of the initial model may be moved in different directions inthe 3D space relative to each other, rather than in the same direction.Furthermore, when multiple three-dimensional vector fields are appliedto the initial model to generate a plurality of perturbed models,corresponding nodes in different perturbed models may be moved indifferent directions in the 3D space relative to each other, rather thanin the same direction. Using a three-dimensional vector field to moveindividual nodes of an initial model in different directions maypreserve the relative structures of the initial model, e.g., such asnetworks of branching faults, while executing random perturbations ofthe remainder of the model. In contrast to conventional systems, inwhich for each perturbed model, a vector having the same direction isapplied to every node, according to embodiments of the invention, adifferent 3D vector having a different direction may be applied to eachdifferent node.

Embodiments of the invention include applying a three-dimensional vectorfield to an initial model. In one embodiment of the invention, applyinga three-dimensional vector field distinguishes applying a group of aplurality of one-dimensional (i.e., unidirectional) vector fields, whereeach of the different one-dimensional vector fields has a differentdirection. For example, when a three-dimensional vector field is appliedto a model, each node of the model may be moved in any differentdirection in 3D space. In contrast, applying a group of a plurality ofone-dimensional vector fields may be similar to partitioning a 3Dmodeled domain and applying a differently directional one-dimensionalvector field to each partition of the 3D modeled domain. Thus, when thevector fields are applied, although nodes in different partitionedregions may be moved in different direction, within any partition, thesame one of the plurality of one-dimensional vector fields is applied,and all the nodes in that region are moved in the same direction inthree-dimensional space.

In an alternative embodiment, the vector fields applied to the initialmodel may be two-dimensional. In this embodiment, when thetwo-dimensional vector field is applied to the initial model, each ofthe nodes of the initial model may be moved in different directions of a2D space, rather than in the same direction.

In one embodiment, the vector field perturbation for each node may berepresented by a function defined on the set of nodes or by a discretecollection of data points. This data may be stored in memory 150 (e.g.in data 155). The data may be, for example, data embedded at each nodeor in an associated matrix, scalar or vector field, lookup data or anyother configuration for storing the data. In another embodiment of theinvention, a gradient of a scalar field may be used instead of thethree-dimensional vector field. The scalar field may have only a singlescalar value, s, stored for each node, which when compared to the threevector values (v_(x),v_(y),v_(z)) stored for a corresponding vectorfield, reduces the effective amount of data in memory 150 (e.g., by ⅔).

In one embodiment, computing system 130 may determine the vector orscalar field perturbation of the model independently for each node.Alternatively, computing system 130 may determine the perturbationstogether for a set or neighborhood of nodes. For example, the scalarfield or the magnitude of the vector field may be averaged or may be amaximum, minimum or other approximation of the individual uncertaintiesof each node in the set. For example, a grouping of individual nodesinto a set of nodes may be based on physical (e.g., spatial)characteristics of the model. For example, the set of nodes may share acell, edge or vertex of a mesh of the model, or the set of nodes mayrepresent a similar geological time, or may be located on the same sideof a fault, etc.

Display 180 may display a visualization of the plurality of perturbedmodels on a graphical user interface. Such visualizations may be forexample stored as data 155. A user may analyze display 180 or initiatecomputations executed by processor 140 to compare and determine thecoherency, consistency, or reliability of each of the models.Alternatively, the user may have knowledge of the terrain and easilyaccept or reject each of the models based on this knowledge. Thegeneration of a plurality of alternative models may be a one-timeprocess or alternatively, may be an iterative process repeated untilfinal results are achieved. For example, if there are multipleacceptable models such that a user cannot narrow his decision to selectto a single model or alternatively, none of the models generated aredesirable, a command may be received, e.g., automatically generated orinput by a user, for computing module to repeat the model generationprocess. In one embodiment, a refinement process may be used. In theexample above, if the user is not satisfied with any of the modelsgenerated by an initial partition or set of points, the user may add orremove model data points, refine a partition, reiterate a process, etc.,to generate a new set of models.

A user may inspect display 180 and remove erroneous data (e.g., bubbles)from the model. For example, if there is a great likelihood thaterroneous data occurs in a specific region (e.g., due to a rapidvariation in the thickness of layers), the user may highlight (e.g.,visually on a displayed model, by selecting points, or by defining thebounds of a region) or otherwise indicate that data may be extractedfrom the region. In one example, the user may send a command to thecomputer module via an input device.

The user may select or highlight a region of the model on display 180 bypointing or dragging a cursor controllable by a mouse on the display ofthe model, entering coordinates of the model, or otherwise indicating amodel location. The user may command computing system 130 to refine aselected region of a model. For example, if a user recognizes a regionof incoherent or inconsistent data (e.g., a region with rapid lateralvariations of layer thicknesses in a sedimentary context known to beunlikely or impossible) the user may highlight or mark that region.Executing a command, computing system 130 may generate various modelsfor each different possible solution corresponding to the uncertainty inthe user selected region, while the remainder of the model may remainunchanged. In the example above, while observing the plurality of modelsgenerated, the user may select only those for which thicknesses areconstant since models with rapid variations of thickness are known to beerroneous. In this or other embodiments, where only a portion of themodel is regenerated for different possible solutions of an uncertainsolution, the boundary of the regenerated and unchanged regions may besmoothed, e.g., by an approximation function.

In another embodiment, the user may set the number of models generated(e.g., 10, 100, or 1,000). For a set range of uncertainty, the greaterthe number of models generated, the more the solution set spans thewhole range of possible solutions.

In one embodiment, input data may include horizon sampling points,including well paths, well markers, or other data. Sampling points maybe displayed on the horizon model. Data removed from the set of samplingpoints (e.g., a pathological set of sampling points forming bubbles) maybe displayed on the model with the retained sampling points, e.g., in adifferent color or by a symbol having a different geometrical shape. Theuser may select normal, reverse or undefined types for a fault byclicking on the name of the fault in a list.

Fault surfaces, horizon surfaces, model boundaries (e.g., as lines ortransparent surfaces) may be modeled and displayed on display 180. Thepaleo-geographic coordinate functions u and v of the uvt-transform maybe “painted” on horizons as a grid (e.g., each line of the grid maycorrespond to a constant value for function u or the function v). Themodel may be displayed in geological space or in geo-chronological spaceor both at the same time.

To model the uncertainties associated with an initial model (horizonsand faults), the initial model and/or one or more perturbed models(horizons and faults) may be displayed on display 180. A neighborhood orvolume around faults for which the fault is uncertain may be displayed.

Reference is again made to FIG. 7, which schematically illustrates agraphical user interface 700 for received input to generate or modifymodels of subsurface structures. Graphical user interface 700 may bedisplayed on display 180, for example, adjacent to the models ofsubsurface structures, or separately from in a preprocessing stage.Graphical user interface 700 may be an interface for requestinginformation from a user (e.g., via input fields, controls for parametersor computer settings, or explicit requests). A user may operate an inputdevice (e.g., input device 165 of FIG. 10), such a mouse or keyboardhaving functionality to alter field, select buttons and otherwiseinteract with graphical user interface 700. Processor 140 may receiveuser input to the graphical user interface 700 and adapt itsfunctionality accordingly. For example, graphical user interface 700 mayprovide data fields 702 and 704 for receiving user-instructions toignore data and use data, respectively, e.g., in modeled region selectedor highlighted by a user. Graphical user interface 700 may provide datafield 706 to reset points that are to be ignored when generating asubsequent model, e.g., so that the user may select or highlight adifferent region. Graphical user interface 700 may provide data field708 to indicate to a user that a computing module is generating horizonsaccording to embodiments of the invention. Graphical user interface 700may provide data field 710 to indicate to a user that one or moregeological properties are being modeled by a computing module. Graphicaluser interface 700 may provide data field 712 to select, e.g., in arange of values, the degree to which data will be refined or filteredwhen generating a model. At one extreme, no data is filtered and atanother extreme, all data is filtered. Graphical user interface 700 mayprovide data field 714 to ignore inconsistent or incoherent data whengenerating a model. For example, a user may select to ignore, filter orremove data points which are further away from a plane which bestapproximates the data by, for example, an amount greater that most otherinput data points (e.g., more than half or another predeterminedpercentage).

Reference is made to FIG. 11, which is a flowchart of a method foridentifying erroneous points in a geological model, which may beperformed for example using system 105 of FIG. 10 or another suitablecomputing system. Other operations or series of operations may be used,and the exact set of steps shown below may be varied.

In operation 1100, a computing device (e.g. processor 140 of FIG. 10)may determine a set of points of a geological model at which ageological-time function has a maximum or minimum value in a set oflocal regions. The local regions may be predefined based on a localpartition of the domain in which the geological model is generated. Forexample, the computing device may define a partition of the model togenerate the set of local regions. Each point in the set of maximum orminimum points may correspond to a different local region in the set oflocal regions.

In operation 1110, a display device (e.g. display 180 of FIG. 10) maydisplay the model on a user interface. The model may include symbolsindicating the location of each point in the set of points at which ageological-time function has the maximum or minimum value.

In operation 1120, the computing device may accept input from the user(e.g., via input device 160 of FIG. 10). User input may indicate one ormore point to be added or removed from the set of points at which ageological-time function has the maximum or minimum value. The user mayalso enter input, which when received by the computing device, mayprevent the symbols indicating the maximum or minimum points from beingdisplayed on the model.

In operation 1130, the computing device may refine the set of pointsaccording to input received from a user for example by adding orremoving points in the set of points at which a geological-time functionhas the maximum or minimum value. The computing device may refine theset of points by refining the partition of the model.

In operation 1140, the display device may display the model to the usergenerated using the refined set of points.

The computing device may accept the data used in the aforementionedoperations as for example a set of data reflected from a subsurfacegeological feature, or such data augmented by another process. Thecomputing device may accept one or more of seismic and well data. Thecomputing device may generate one or more of seismic and well data.

Reference is made to FIG. 12, which is a flowchart of a method forrefining a first geological model, which may be performed for exampleusing system 105 of FIG. 10 or another suitable computing system. Otheroperations or series of operations may be used, and the exact set ofsteps shown below may be varied.

In operation 1200, a computing device (e.g. processor 140 of FIG. 10)may identify a first set of nodes at which a geological-time functionhas a local maximum or minimum value in a first geological model.

In operation 1210, the computing device may generate a second set ofnodes. The second set of nodes may correspond to the first set of nodes,where values of the geological-time function at the second set of nodesare substantially different than the values of the geological-timefunction at the first set of nodes. The values of the geological-timefunction at the second set of nodes are typically not a local maximum orminimum value. The computing device may apply a smoothness constraint tothe values of the geological-time function at the second set of nodes.The computing device may approximate the values of the second set ofnodes from values of nodes in the first geological model that neighborthe replaced first set of nodes.

In operation 1220, the computing device may replace the values at thefirst set of nodes in the first geological model with the values at thesecond set of nodes.

In operation 1230, a display device (e.g. display 180 of FIG. 10) maydisplay the first geological model in which the values of the first setof nodes are replaced with the values at the second set of nodes. Thedisplay may optionally display the first set of nodes at which ageological-time function has a local maximum or minimum value.

The computing device may accept the data used in the aforementionedoperations as for example a set of data reflected from a subsurfacegeological feature, or such data augmented by another process. Thecomputing device may accept one or more of seismic and well data. Thecomputing device may generate one or more of seismic and well data.

Reference is made to FIG. 13, which is a flowchart of a method forgenerating a model by defining a combination of geological properties ofthe model, which may be performed for example using system 105 of FIG.10 or another suitable computing system. Other operations or series ofoperations may be used, and the exact set of steps shown below may bevaried.

In operation 1300, a computing device (e.g. processor 140 of FIG. 10)may define, for each node of the model, a value for each of a pluralityof distinct geological properties associated with the model. Thegeological properties may include for example the geometry of faults,geometry of well paths, geometry of horizons, the occurrence of samplingpoints and seismic travel times.

In operation 1310, the computing device may generate a linearcombination of the values of the plurality of geological properties foreach node of the model. The computing device may accept data from a userfor defining values of a geological property associated with one or morenodes of the model.

In operation 1320, the computing device may calculate a geological-timefunction at each node of the model using the linear combination of thevalues of the geological properties associated with the node of themodel. The values of each of the plurality of distinct geologicalproperties may be simultaneously used as a linear combination togenerate the geological-time function. Accordingly, the model generatedusing the linear combination may have the plurality of geologicalproperties corresponding thereto at each node of the model.

In operation 1330, a display device (e.g. display 180 of FIG. 10) maydisplay the model.

In one embodiment, the model may be generated using control-point data,wherein control-points derived from seismic cross sections or seismiccubes and control-points observed on well paths are used to generatesubstantially separate horizons.

The computing device may accept the data used in the aforementionedoperations as for example a set of data reflected from a subsurfacegeological feature, or such data augmented by another process. Thecomputing device may accept one or more of seismic and well data. Thecomputing device may generate one or more of seismic and well data.

Reference is made to FIG. 14, which is a flowchart of a method forperturbing an initial three-dimensional (3D) geological model using a 3Dvector field, which may be performed using system 105 of FIG. 10. Otheroperations or series of operations may be used, and the exact set ofsteps shown below may be varied.

In operation 1400, a computing device (e.g. processor 140 of FIG. 10)may generate a 3D vector field including 3D vectors. For example, acoherent vector field may be generated. Each 3D vector of the 3D vectorfield may be associated with a node of the initial 3D geological modeland may have a magnitude within a range of uncertainty of the node ofthe initial 3D geological model associated therewith. In one embodiment,the uncertainty at the nodes of the model may correspond to the locationof the nodes. In another embodiment, the uncertainty at the nodes of themodel may correspond to the location of a fault. In yet anotherembodiment, the uncertainty at the nodes of the model may beproportional to the wavelength of data reflected from a subsurfacegeological feature.

The 3D geological model has three dimensions, and the 3D vector fieldassociated with each node of the 3D geological model may be defined bythree values, each of which corresponds to a different one of the threedimensions of the 3D geological model. In one embodiment, the vectorfield is defined by a 3D gradient of a scalar field. In this embodiment,a scalar value may be associated with each node of the geological modeland the 3D vector field may be generated by taking a three dimensionalgradient of the scalar field.

In operation 1410, the computing device may apply the 3D vector field tothe initial 3D geological model associated therewith to generate analternative 3D model. The alternative 3D model may differ from theinitial 3D geological model by a 3D vector at the nodes having uncertainvalues. The computing device may apply separately one or more additionaldifferent 3D vector fields to the initial 3D geological model togenerate a plurality of alternative 3D models. In one embodiment, thecomputing device may represent the 3D vector field by a first matrix andthe initial 3D geological model by a second matrix. In this embodiment,the computing device may apply the 3D vector field to the initial 3Dgeological model by adding the first and second matrices together.

In operation 1420, a display device (e.g. display 180 of FIG. 10) maydisplay the alternative 3D model.

The computing device may accept the data used in the aforementionedoperations as for example a set of data reflected from a subsurfacegeological feature, or such data augmented by another process. Thecomputing device may accept one or more of seismic and well data. Thecomputing device may generate one or more of seismic and well data.

Reference is made to FIG. 15, which is a flowchart of a method for, in athree-dimensional (3D) subsurface model, rescaling a function of time ofwhen a geological structure was originally formed in the Earth, whichmay be performed using for example system 105 of FIG. 10 or anothersuitable computing system. Other operations or series of operations maybe used, and the exact set of steps shown below may be varied.

In operation 1500, a computing device (e.g. processor 140 of FIG. 10)may generate a monotonically increasing function. The computing devicemay automatically generate the monotonically increasing function, forexample, to be a random function.

In operation 1510, the computing device may generate a rescaledgeological time function by applying a rescaling function to themonotonically increasing function. The rescaling function may be definedso that the norm of the gradient of the rescaled geological-timefunction is, on average, approximately constant throughout the model.The computing device may automatically generate the constant value ofthe norm of the gradient of the rescaled geological-time function to be,for example one (1). A computer memory unit (e.g. memory 150 of FIG. 10)may store the rescaled geological-time function.

In operation 1520, a display device (e.g. display 180 of FIG. 10) maydisplay a three-dimensional model of the structure when it wasoriginally formed, the three-dimensional model defined by the rescaledgeological time function in one of the three dimensions of the model.

The computing device may accept the data used in the aforementionedoperations as for example a set of data reflected from a subsurfacegeological feature, or such data augmented by another process. Thecomputing device may accept one or more of seismic and well data. Thecomputing device may generate one or more of seismic and well data.

Reference is made to FIG. 16, which is a flowchart of a method formodeling the probability of leakage across a fault in a model, which maybe performed using for example system 105 of FIG. 10 or another suitablecomputing system. Other operations or series of operations may be used,and the exact set of steps shown below may be varied.

In operation 1600, a computing device (e.g. processor 140 of FIG. 10)may, for each fault node associated with the modeled fault, identify twoor more nodes that are substantially adjacent to the fault node and thatare located on opposite sides of the fault from each other.

In operation 1610, the computing device may identify a region of themodel representing a subsurface structure having hydrocarbons.

In operation 1620, the computing device may determine the probability ofleakage across the modeled fault for each fault node. The probability ofleakage may be determined based on whether or not the two or more nodesidentified in operation 1600 are determined to be located in the regionof the model identified in operation 1610. The computing device mayautomatically determine if the two or more nodes are located in theidentified region. Alternatively or additionally, the computing devicemay receive data manually entered by a user (e.g., input device 165 ofFIG. 10) indicating if the two or more nodes are located in theidentified region. In one embodiment, the computing device mayinitialize a counter for each node and may increase the probabilityassociated with each node by incrementing the counter by a value, e.g.,one, if the two or more identified nodes are determined to be located inthe identified region of the model.

In operation 1630, a computer memory (e.g. memory 150 of FIG. 10) maystore the probability of leakage associated with each fault node. Foreach fault node, the probability of leakage may be greater when the twoor more nodes are determined to be located in the identified region ofthe model than if the two or more nodes are determined to be locatedoutside the identified region. The computing device may generate a totalprobability of leakage across the modeled fault by averaging theprobabilities for each fault node.

In one embodiment, the computing device may repeat operations 1600-1630for a node in each of a plurality of equiprobable models that correspondto the fault node. In this embodiment, the probability of leakage foreach node may be an average of the probabilities of leakage for allcorresponding nodes of the plurality of equiprobable models.

In operation 1640, a display device (e.g. display 180 of FIG. 10) maydisplay the model and the probability of leakage associated with eachfault node using a color scale overlaying the model.

The computing device may accept the data used in the aforementionedoperations as for example a set of data reflected from a subsurfacegeological feature, or such data augmented by another process. Thecomputing device may accept one or more of seismic and well data. Thecomputing device may generate one or more of seismic and well data.

When used herein, “generating” or “creating” a model or a subsurfacefeature may include creating data representing the model or feature, thedata being stored for example in a computer system such as that shown inFIG. 10.

In the foregoing description, various aspects of the present inventionhave been described. For purposes of explanation, specificconfigurations and details have been set forth in order to provide athorough understanding of the present invention. However, it will alsobe apparent to one skilled in the art that the present invention may bepracticed without the specific details presented herein. Furthermore,well known features may have been omitted or simplified in order not toobscure the present invention. Unless specifically stated otherwise, asapparent from the following discussions, it is appreciated thatthroughout the specification discussions utilizing terms such as“processing,” “computing,” “calculating,” “determining,” or the like,refer to the action and/or processes of a computer or computing system,or similar electronic computing device, that manipulates and/ortransforms data represented as physical, such as electronic, quantitieswithin the computing system's registers and/or memories into other datasimilarly represented as physical quantities within the computingsystem's memories, registers or other such information storage,transmission or display devices. In addition, the term “plurality” maybe used throughout the specification to describe two or more components,devices, elements, parameters and the like.

Embodiments of the invention may manipulate data representations ofreal-world objects and entities such as underground geological features,including faults and other features. Data received by for example areceiver receiving waves generated by an air gun or explosives may bemanipulated and stored, e.g., in memory 150, and data such as imagesrepresenting underground features may be presented to a user, e.g., as avisualization on display 180.

When used herein, geological features such as horizons and faults mayrefer to the actual geological feature existing in the real world, orcomputer data representing such features (e.g., stored in a memory ormass storage device). Some features when represented in a computingdevice may be approximations or estimates of a real world feature, or avirtual or idealized feature, such as an idealized horizon as producedin a uvt-transform. A model, or a model representing subsurface featuresor the location of those features, is typically an estimate or a“model”, which may approximate or estimate the physical subsurfacestructure being modeled with more or less accuracy.

It should be recognized that embodiments of the present invention maysolve one or more of the objectives and/or challenges described in thebackground, and that embodiments of the invention need not meet everyone of the above objectives and/or challenges to come within the scopeof the present invention. While certain features of the invention havebeen particularly illustrated and described herein, many modifications,substitutions, changes, and equivalents may occur to those of ordinaryskill in the art. It is, therefore, to be understood that the appendedclaims are intended to cover all such modifications and changes in formand details as fall within the true spirit of the invention.

What is claimed is:
 1. A method for perturbing an initialthree-dimensional (3D) geological model including a fault using a 3Dvector field, the method being executed by a processor according tosoftware instructions, the method comprising: using the processor,generating a coherent 3D vector field comprising 3D vectors, whereineach 3D vector of the 3D vector field is associated with a node of theinitial 3D geological model and has a magnitude within a range ofuncertainty of the node of the initial 3D geological model associatedtherewith, wherein a plurality of the 3D vectors of the 3D vector fieldhave different directions in a 3D space; and using the processor,applying the coherent 3D vector field to the initial 3D geological modelincluding the fault to generate a perturbed 3D model, wherein the faultis a non-vertical fault, wherein the perturbed 3D model differs from theinitial 3D geological model by a displacement in the differentdirections in the 3D space defined by the 3D vector field associatedwith nodes having uncertain values; and displaying the perturbed 3Dmodel.
 2. The method of claim 1, comprising applying separately one ormore additional different 3D vector fields to the initial 3D geologicalmodel to generate a plurality of perturbed 3D models.
 3. The method ofclaim 1, wherein the 3D vector field is coherent such that thedivergence of the 3D vector field is greater than (−1).
 4. The method ofclaim 1, wherein the 3D vector field is coherent such that, for anyinfinitely small polyhedral volume having vertices and faces in theinitial 3D geological model, the vertices are never displaced across anyof the polyhedron faces.
 5. The method of claim 1, wherein the 3D vectorfield is equal to the gradient of a 3D scalar field.
 6. The method ofclaim 1, comprising representing the 3D vector field at the nodes of afirst mesh and representing the initial 3D geological model by a secondmesh, wherein applying the 3D vector field to the initial 3D geologicalmodel comprises moving the nodes of the second mesh.
 7. The method ofclaim 1, wherein the 3D geological model has three dimensions, andwherein the 3D vector field associated with each node of the 3Dgeological model is defined by three values, each of which correspondsto a different one of the three dimensions of the 3D geological model.8. The method of claim 1, wherein the uncertainty at the nodes of theinitial 3D geological model corresponds to uncertainty of a location ofa fault.
 9. The method of claim 1, wherein the uncertainty at the nodesof the initial 3D geological model corresponds to uncertainty of aseismic velocity field.
 10. The method of claim 1, wherein theuncertainty at the nodes is proportional to the wavelength of datareflected from a subsurface geological feature.
 11. The method of claim1, comprising accepting a set of data reflected from a subsurfacegeological feature, wherein the uncertainty values of the nodes aregenerated using the set of reflected data.
 12. The method of claim 1,wherein the vector field is a combination of two or more distinct vectorfields, each of which is associated with a different type ofuncertainty.
 13. The method of claim 1, wherein each of the nodes of theinitial 3D geological model is associated with a first parameterproportional to geological time, the method comprising perturbing a setof the nodes by associating a second parameter with each node, whereinthe derivative of the second parameter relative to the first parameteris greater than (−1).
 14. A system configured to perturb an initialthree-dimensional (3D) geological model including a fault using a 3Dvector field, the system comprising: a memory to store the initial 3Dgeological model; a processor configured to generate a coherent 3Dvector field comprising 3D vectors, wherein each 3D vector of the 3Dvector field is associated with a node of the initial 3D geologicalmodel and has a magnitude within a range of uncertainty of the node ofthe initial 3D geological model associated therewith, wherein aplurality of the 3D vectors of the 3D vector field have differentdirections in a 3D space, and to apply the coherent 3D vector field tothe initial 3D geological model including the fault to generate aperturbed 3D model, wherein the fault is a non-vertical fault, whereinthe perturbed 3D model differs from the initial 3D geological model by adisplacement in the different directions in the 3D space defined by the3D vector field associated with nodes having uncertain values; and adisplay to display the perturbed 3D model.
 15. The system of claim 14,wherein the processor is configured to separately apply one or moreadditional different 3D vector fields to the initial 3D geological modelto generate a plurality of perturbed 3D models.
 16. The system of claim14, wherein the processor is configured to generate the 3D vector fieldto be coherent such that the divergence of the 3D vector field isgreater than (−1).
 17. The system of claim 14, wherein the processor isconfigured to generate the 3D vector field to be coherent such that, forany infinitely small polyhedral volume having vertices and faces in theinitial 3D geological model, the vertices are never displaced across anyof the polyhedron faces.
 18. The system of claim 14, wherein theprocessor is configured to generate the 3D vector field to be equal tothe gradient of a 3D scalar field.
 19. The system of claim 14, whereinthe processor is configured to represent the 3D vector field at thenodes of a first mesh and represent the initial 3D geological model by asecond mesh, wherein the processor displaces the nodes of a second meshby applying the 3D vector field to the initial 3D geological model. 20.The system of claim 14, wherein the processor is configured to generatethe initial 3D geological model to have three dimensions and to generatethe 3D vector field associated with each node of the 3D geological modelto be defined by three values, each of which corresponds to a differentone of the three dimensions of the 3D geological model.
 21. The systemof claim 14, comprising a receiving unit configured to receive a set ofdata reflected from a subsurface geological feature, wherein thereflected set of data is used to model the subsurface geological featurein the initial 3D geological model.
 22. The system of claim 14, whereinthe processor is configured to generate the vector field to be acombination of two or more distinct vector fields, each of which isassociated with a different type of uncertainty.
 23. The system of claim14, wherein the processor is configured to associate each of the nodesof the initial 3D geological model with a first parameter proportionalto geological time and to perturb a set of the nodes by associating asecond parameter with each node, wherein the derivative of the secondparameter relative to the first parameter is greater than (−1).